Unit 2 Chapter 6 – Quadratic Functions 1
6.1 – Exploring Quadratic Relations
(I) Review of Functions
What is a function?
Using Function Notation P(h) =
Goals:
Review of Functions
What is a Quadratic Function?
Exploring Quadratic Relations in Standard
Form and Changing Coefficients
Unit 2 Chapter 6 – Quadratic Functions 2
A function
can algebraically model data for the purpose of prediction
(ie. extrapolate beyond data)
cannot produce two output values (range values) for the same
input value (domain value CANNOT REPEAT.)
Example: Which graphs below represent a function?
1. 2.
(II) What is a Quadratic Function?
In each case, sketch the trajectory of each object.
(a) (b)
The _______________________ test
graphically determines which relations are
functions.
height
time
height height height
time
Unit 2 Chapter 6 – Quadratic Functions 3
The shape of a quadratic relation is known as a ___________________.
Quadratic functions: represent the trajectory a ball makes when thrown.
The path a ball travels gives a special “U” shape called a “parabola.”
This parabolic shape occurs in many natural phenomena such as kicking a
football/soccer ball, hitting a golf ball, flight path of birds, parabolic satellites, etc.
The parabolic shape is the result of sketching Quadratic Functions.
What are Quadratic Functions?
The simplest quadratic function is y = x2.
(The word quadratic comes from the word quadratum, a Latin word meaning
square.)
Quadratic Functions are the result of multiplying two linear functions:
For example, expand: y = (x + 1)(x + 4)
Note: The degree of a quadratic
function (the highest
exponent on a term). Degree = ______
Unit 2 Chapter 6 – Quadratic Functions 4
Which of the functions are quadratic ( remember: degree must be 2)?
i) y = 5(x + 3) ii) y = 5x(x + 3)
iii) y = 5(x2 + 3) iv) y = (5x + 1)(x + 3)
v) y = 5x(x2 + 3) vi) y = 5(x + 3)
2 + 1
The simplest form of a quadratic function is y = x2.
Sketch the graph by producing a set of points.
x y = x2
The parabola is symmetric about a line called the axis of symmetry.
This lines divides the graph into two equal parts
It is a mirror image
It intersects the parabola at the vertex
Where is the axis of symmetry located above?
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 2
- 1
1
2
3
4
5
Unit 2 Chapter 6 – Quadratic Functions 5
A Quadratic Relation
can be written in standard form as y = ax2 + bx + c (where a 0).
(III) Exploring the graph of a quadratic relation in standard form
y = ax2 + bx + c based on changing the coefficients (a, b and c)
Sketch the graph of each quadratic relation based on the values of the coefficients.
(Check out https://www.desmos.com/)
(A) Varying ‘a’ while ‘b’ and ‘c’ are constant.
When b = 0 and c = 0. Draw the line of symmetry in each graph.
(i) a = 2 (ii) a =
Eqn in Standard Form:___________ Eqn in Standard Form:___________
(iii) a = –3
Eqn in Standard Form:___________
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 2
- 1
1
2
3
4
5
6
7
8
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 5
- 4
- 3
- 2
- 1
1
2
3
4
5
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 12- 11- 10
- 9- 8- 7- 6- 5- 4- 3- 2- 1
1 How did the graph change as ‘a’ varied?
Unit 2 Chapter 6 – Quadratic Functions 6
NOTE: •The size of ‘a’ dictates the width of opening
•The sign of ‘a’ indicates DIRECTION OF OPENING
(i) If a is positive (a > 0) then the graph opens _________.
(ii) If a is negative (a < 0) then the graph opens _________.
Summary:
Standard Form y = ax2 + bx + c
Degree of all quadratic functions is 2 (largest exponent)
A quadratic function is a parabola with a vertical line of symmetry
The highest or lowest point lies on the line of symmetry
If ‘a’ is positive graph opens up and opens down if ‘a’ is negative
Practice Questions: P.324 #1, #2, #5
Unit 2 Chapter 6 – Quadratic Functions 7
6.2 – Properties of Graphs of Quadratic Functions
(I) Using symmetry to estimate the coordinates of a point
Example: The height of a volleyball against time
(i) At what time did the volleyball reach
its greatest height?
t0.5 1 1.5 2
h
2
4
6
8
10
Time (s)
He
ig
ht (ft)
Goals:
Using Symmetry to Estimate the Coordinates of a Point
What is a Quadratic Function?
Characteristics of a Quadratic Function, Vertex, Axis of
Symmetry & Max/Min values
Attaining the Vertex of a Quadratic Function
Unit 2 Chapter 6 – Quadratic Functions 8
(ii) Use the graph to approximate the greatest height?
(iii) Using the answers to (i) and (ii), what are the coordinates
of the highest point?
Vertex:
The point at which the quadratic
function reaches its maximum
or minimum value.
Maximum Value
When the graph opens down, the
vertex is the ________ point on
the graph and the y-coordinate of
the vertex is the ____________
value.
Minimum Value
•When the graph opens up the vertex is
the ________ point on the graph and the
y-coordinate is the _____________ value.
The value of ‘a’ in a quadratic function
y = ax2 + bx + c determines whether the
vertex is a maximum or minimum point.
Unit 2 Chapter 6 – Quadratic Functions 9
(II) Attaining a maximum or minimum value through
a table of values or graph
Example:
A golf ball is struck and its height with respect to time is represented by
the function h(t) = –3t2 + 12t where h(t) represents height and t is the time
in seconds.
(a) What’s the direction of opening?
(b) Will the ball attain a maximum or minimum height?
(c) What’s the y – intercept?
(d) Create a table of values and graph the function.
t h(t)
(e) How long does it take for the ball to attain its maximum height?
(f) What is the maximum height attained?
(g) What are the coordinates of the vertex?
t1 2 3 4
h
1
2
3
4
5
6
7
8
9
10
11
12
Time (s)
He
ig
ht (ft)
Unit 2 Chapter 6 – Quadratic Functions 10
(III) Attaining a maximum or minimum value through a
graph and formula
For each function indicated in the table determine:
(i) the vertex (using graphing software https://www.desmos.com/)
(ii) the equation of axis of symmetry
(iii) the values of ‘a’, ‘b’ and ‘c’ from the function y = ax2 + bx + c
(iv) the value of
Function
y = ax2 + bx + c
Vertex Equation of Axis
of Symmetry
a b c
y = x2 – 4x + 7
y = –2x2 – 4x + 7
y = 3x2 – 6x + 10
(a) What do you notice about the x– coordinate of each vertex, the
equation of axis of symmetry and the value of
?
(b) Once the x– coordinate of the vertex is attained from a quadratic
function such as y = –2x2 – 4x + 7, how could we algebraically attain
the y– coordinate?
Unit 2 Chapter 6 – Quadratic Functions 11
The vertex and equation of axis of symmetry of a quadratic function
can be attained:
(A) Graphically
Example: State the vertex and equation of axis of symmetry.
Vertex:__________
Equation of axis of symmetry:______
(B) Tabulation
Example: State the vertex and equation of axis of symmetry.
Vertex:__________ Equation of axis of symmetry:______
Summary Attaining the vertex of a quadratic function y = ax2 + bx + c
(i) Get the x– coordinate of the vertex by the formula x =
(ii) Substitute that result into y = ax2 + bx + c to attain the
y– coordinate.
Unit 2 Chapter 6 – Quadratic Functions 12
(C) Algebraically
Example: Determine the vertex and equation of axis of symmetry
for y = 2x2 – 8x + 7.
(IV) Determining the axis of symmetry from a set of points.
Example: Determine the equation of axis of symmetry from the parabola.
Where is the line of symmetry positioned compared to the
location of the two given points?
P.333 – P.334 #2 Determine y – intercept #6 #11b
Unit 2 Chapter 6 – Quadratic Functions 13
Example: Determine the equation of axis of symmetry for each parabola
that contains the points:
(a) (–2, 4) and (6, 4) (b) (5, 0) and (11, 0)
(V) Attaining the domain and range of a quadratic function
Review of Domain and Range
Domain
is the set of all input values (or x–values)
Range
is the set of all output values (or y–values)
The domain and range can be attained:
(i) Graphically
(ii) Tabulation (or set of points)
(iii) Function
Summary Axis of symmetry
(i) A vertical reflection line that passes through the vertex
(ii) Can be attained by the formula x =
when the quadratic
function y = ax2 + bx + c is given.
(iii) Can be attained from two points with the same y–coordinate
by averaging the x–coordinates.
Unit 2 Chapter 6 – Quadratic Functions 14
Determining Domain & Range Graphically
1. State the domain using set notation for:
(a) (b)
Set notation:________________ Set notation:________________
(c) (d)
Set notation:________________ Set notation:________________
2. State the range using set notation for:
(a) (b)
Set notation:________________ Set notation:________________
-4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3
-4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3
- 4
- 3
- 2
- 1
0
1
- 4
- 3
- 2
- 1
0
1
Unit 2 Chapter 6 – Quadratic Functions 15
3. State the domain and range for:
(a) (b)
Domain:__________________ Domain:__________________
Range:___________________ Range:____________________
NOTE: For Domain - all points on the graph FALL to the x-axis
For Range - all points on the graph MOVE OVER to the y-axis
x-6 -5 -4 -3 -2 -1 1 2 3 4
y
-5-4-3-2-1
12345
x-6 -5 -4 -3 -2 -1 1 2 3 4
y
-5-4-3-2-1
12345
Unit 2 Chapter 6 – Quadratic Functions 16
Determining Domain & Range from a Quadratic Function
How do we attain the domain of a quadratic function such as y = –2x2 + 4x + 1
without the aid of a graph?
(a) What is the direction of opening for the given function?
(b) Will the function have a maximum or minimum value?
(c) How can we algebraically attain the maximum/minimum value?
(d) How does the above information enable us to express the range?
SUMMARY: To attain the domain and range from y = ax2 + bx + c
Domain – for any quadratic function is _____________
Range (i) determine the _____________ of opening
(ii) determine the ______________of vertex by ________
(iii) Substitute the result from (ii) into the function
________________to get the maximum/minimum value
(iv) State the range.
Unit 2 Chapter 6 – Quadratic Functions 17
Example: Determine the domain and range for:
(a) y = 3x2 – 2 (b) y = x
2 + 4x + 4
(c) y = –x2 + 6x – 8
P.333 – P.335 #4, #5, #7(a) (i) (b), (c) #9b, c #10 #13a, b #14
Unit 2 Chapter 6 – Quadratic Functions 18
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
SKETCHING GRAPHS OF QUADRATIC FUNCTIONS
IN STANDARD FORM
EXAMPLE 1
Given the function y = 4x2 + 8x + 6 determine the characteristics and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
Number of x–intercepts:
Y–intercept:
Domain:
Range:
Unit 2 Chapter 6 – Quadratic Functions 19
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
EXAMPLE 2
Given the function y = –3x2 + 12x – 7 determine the characteristics and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
Number of x–intercepts:
Y–intercept:
Domain:
Range:
Unit 2 Chapter 6 – Quadratic Functions 20
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
PRACTICE QUESTIONS
1. Given the function y = 2x2 – 12x + 8 determine the characteristics and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
Number of x–intercepts:
Y–intercept:
Domain:
Range:
2. Given the function y = –2x2 + 8x – 10 determine the characteristics and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
Number of x–intercepts:
Y–intercept:
Domain:
Range:
Unit 2 Chapter 6 – Quadratic Functions 21
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
3. Given the function y = –3x2 – 6x – 7 determine the characteristics and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
Number of x–intercepts:
Y–intercept:
Domain:
Range:
4. Given the function y = x2 + 6x + 9 determine the characteristics and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
Number of x–intercepts:
Y–intercept:
Domain:
Range:
Unit 2 Chapter 6 – Quadratic Functions 22
6.3 – Factored Form of a Quadratic Function (Part I)
(I) Identifying characteristics from the graph of a quadratic function
Example: Given the graph of a quadratic function ,
state and label in the graph:
(a) the coordinates of the vertex.
(b) the axis of symmetry
(state the equation)
(c) the y – intercept.
(state the coordinates)
(d) the x – intercepts.
(state the coordinates)
x-1 1 2 3 4 5 6 7 8
y
-6-4-2
2468
101214
Note: x – intercepts
When we state the coordinates of the
x – intercepts, the y – coordinate is
always ______.
Goals:
Identifying Characteristics from a Graph of a Quadratic Function
(Vertex, Axis of Symmetry, y – intercept and x – intercept)
Algebraically Attaining Characteristics from a Quadratic Function
in Standard Form
Algebraically Attaining Characteristics from a Quadratic Function
in Factored Form
Unit 2 Chapter 6 – Quadratic Functions 23
(II) Determining characteristics of a quadratic function from the equation.
Example: For the quadratic function y = x2 – 8x + 12
determine:
(a) the coordinates of the y – intercept.
(b) the equation of axis of symmetry.
(c) the coordinates of the vertex
How can we determine each of the characteristics noted above without
the aid of a graph?
Attaining the vertex of a quadratic
function y = ax2 + bx + c
(i) Get the x– coordinate of the vertex (or
equation of axis of symmetry)
by the formula x =
(ii) Substitute that result into
y = ax2 + bx + c to attain the y– coordinate.
How can we algebraically determine the x – intercepts from the
quadratic function y = x2 – 8x + 12 ?
Unit 2 Chapter 6 – Quadratic Functions 24
(III) Investigating how to algebraically attain x – intercepts from a
quadratic function in standard form y = ax2 + bx + c.
Graph (https://www.desmos.com/) each of the quadratic functions and
(i) state the coordinates of the vertex
(ii) equation of axis of symmetry
(iii) state the coordinates of the y – intercept
(iv) coordinates of the x – intercepts
(i) y = x2 – 8x + 12 (ii) y = (x – 2)(x – 6)
Coordinates of vertex:_________ Coordinates of vertex:_________
Axis of symmetry:__________ Axis of symmetry:__________
Coordinates of y – intercept:_______ Coordinates of y – intercept:_______
Coordinates of x – intercepts:_______ Coordinates of x – intercepts:_______
x-1 1 2 3 4 5 6 7
y
-6
-4
-2
2
4
6
8
10
12
x-1 1 2 3 4 5 6 7
y
-6
-4
-2
2
4
6
8
10
12
Unit 2 Chapter 6 – Quadratic Functions 25
(iii) y = 2x2 – 4x – 6 (iv) y = 2(x + 1)(x – 3)
Coordinates of vertex:_________ Coordinates of vertex:_________
Axis of symmetry:__________ Axis of symmetry:__________
Coordinates of y – intercept:__________ Coordinates of y – intercept:________
Coordinates of x – intercepts:___________ Coordinates of x – intercepts:________
Using the results of the quadratic functions and their respective graphs, answer
each of the questions.
1. Which form of the quadratic function is easiest for determining the
x – intercepts without the graph?
x-2 -1 1 2 3 4 5
y
-8
-6
-4
-2
2
Note: Forms of a Quadratic Function
(i) Standard Form y = ax2 + bx + c
(ii) Factored Form y = a(x – r)(x – s)
x-2 -1 1 2 3 4 5
y
-8
-6
-4
-2
2
Unit 2 Chapter 6 – Quadratic Functions 26
2. How can we attain the x – intercepts by looking at the quadratic function?
3. What is the connection between the factored form of a quadratic function
such as, y = (x – 2)(x – 6) and the x – intercepts?
4. What is the value of the y – coordinate at the point where the graph crosses
the x – axis?
5. How can we algebraically determine the x – intercepts of a quadratic
function such as y = x2 – 8x + 12 ?
Zero Product Property
If the product of two real numbers
is zero ( a • b = 0) then one or both
must be zero.
In other words: a = 0 and b = 0
x
y
Zeros of a Quadratic Function
●The zero(s) of a quadratic function represent
the position(s) where the height is _________
●The zero(s) of a quadratic function
are also referred to as the _________________
Unit 2 Chapter 6 – Quadratic Functions 27
(IV) Algebraically determining characteristics of a quadratic function
expressed in Factored Form y = a(x – r)(x – s).
Example 1: Given the quadratic function y = –( x + 2)(x – 4)
(a) determine the x – intercepts
(b) determine the axis of symmetry
(c) determine the coordinates of the vertex
(d) determine the y – intercept
(e) sketch the graph
(f) state the range
x-3 -2 -1 1 2 3 4 5 6
y
-2
2
4
6
8
10
Unit 2 Chapter 6 – Quadratic Functions 28
Example 2: Given the quadratic function y = 2x( x + 4)
(a) determine the x – intercepts
(b) determine the axis of symmetry
(c) determine the coordinates of the vertex
(d) determine the y – intercept
(e) sketch the graph
(f) state the range
x-6 -5 -4 -3 -2 -1 1 2
y
-10
-8
-6
-4
-2
2
Unit 2 Chapter 6 – Quadratic Functions 29
Example 3: Given the quadratic function y = –3(x – 1)(x – 1)
(a) determine the x – intercepts
(b) determine the axis of symmetry
(c) determine the coordinates of the vertex
(d) determine the y – intercept
(e) sketch the graph
(f) state the range
x-2 -1 1 2 3 4
y
-6
-4
-2
2
4
P. 346 – P.347
#1(a) – (f), #2(a) – (c), #4(a), (d), (e), #5
Unit 2 Chapter 6 – Quadratic Functions 30
6.3 – Factored Form of a Quadratic Function (Part II)
(I) Identifying characteristics from the graph to determine the equation
of a Quadratic Function
Example: Determine the function that defines this parabola. Express
the function in standard form y = ax2 + bx + c.
x-4 -3 -2 -1 1 2 3 4 5 6
y
-6-4-2
2468
101214161820
Step I:
•Using the x – intercepts express the
function in factored form.
Step II:
•Use the coordinates of another
point to solve for the value of ‘a’.
Step III:
•Expand factored form to produce
standard form.
Goals:
Identifying Characteristics from a Graph to Determine the
Equation of a Quadratic Function
Determining the Equation of a Quadratic Function based on a
Verbal Description
Unit 2 Chapter 6 – Quadratic Functions 31
Example: Determine the function that defines this parabola. Express
the function in standard form y = ax2 + bx + c.
Step I:
•Using the x – intercepts express the
function in factored form.
Step II:
•Use the coordinates of another
point to solve for the value of ‘a’.
Step III:
•Expand factored form to produce
standard form.
x-2 -1 1 2 3 4 5 6 7 8
y
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
2
Unit 2 Chapter 6 – Quadratic Functions 32
(II) Determining the equation of a Quadratic Function based on
a verbal description.
Example: A missile fired from ground level attains a height of 180 m
at 2 seconds. The missile is in the air for 6 seconds
(a) Determine the quadratic function that models the height
of the missile over time.
(b) State the domain and range of the variables.
Domain:_________________
Range:__________________
t
h(t)
P.348 – 349 #11 (a – d), #13, #15, #17
Unit 2 Chapter 6 – Quadratic Functions 33
6.3 – Factored Form of a Quadratic Function (Part III)
Maximum/Minimum Word Problems
To solve Max/Min Problems you will have to determine the highest (or lowest )
point , in other words, the vertex.
Goals:
Solving Problems of Trajectory that Model Quadratic Functions
Solving Application Problems (Area and Revenue) that Model
Quadratic Functions
x
y
x
y
To determine when the minimum or
maximum value occurs use:
To determine the maximum or minimum value
substitute
into y = ax2 + bx + c.
Unit 2 Chapter 6 – Quadratic Functions 34
(I) Solving Problems of Trajectory that Model Quadratic Functions
Example 1: A toy rocket is fired into the air, from the ground, and its height
h(t) above the ground in meters, after t seconds, is modeled by
the function h(t) = –t2 + 24t.
(a) What is the initial height of the rocket?
(b) What was the height of the ball at 5 seconds?
(c) When did the ball reach its maximum height?
(d) What was the maximum height of the ball?
(e) What is the domain and the range of the variables in the function?
Example 2: A flare is fired into the air and its height h(t) above the ground, in meters,
after t seconds, is modeled by the function h(t) = –2t2 + 16t + 24.
(a) What is the initial height of the rocket?
(b) What was the height of the ball at 2 seconds?
(c) When did the ball reach its maximum height?
(d) What was the maximum height of the ball?
(e) What is the domain and the range of the variables in the function?
Unit 2 Chapter 6 – Quadratic Functions 35
(II) Solving Area Problems that Model Quadratic Functions
Example 1.
A rectangular play enclosure for some dogs is to be made with 60 m of
fencing using the kennel as one side of the enclosure as shown. The
quadratic function that models the area of the enclosure is represented by
function A(x) = –2x2 + 60x where A(x) represents the area enclosed and x
represents the width in meters.
(a) Determine the maximum area.
(b) State the domain and range of the variables in the function.
Kennel
Play Enclo
sure
x x
60 – 2x
Unit 2 Chapter 6 – Quadratic Functions 36
Example 2.
A rectangular region, placed against the wall of a house, is divided into
three regions of equal area using a total of 120 m of fencing as shown.
The quadratic function that models the area of the enclosure is represented
by function A(x) = –4x2 + 120x where A(x) represents the area enclosed and
x represents the width in meters.
(a) Determine the maximum area.
(b) State the domain and range of the variables in the function.
fencingWall of House
x x x x
120 – 4x
Unit 2 Chapter 6 – Quadratic Functions 37
Extra Practice Problems:
1. A toy rocket is fired into the air, from the ground, and its height h(t) above the
ground in meters, after t seconds, is modeled by the function h(t) = –t2 + 16t.
(a) What is the initial height of the rocket?
(b) What was the height of the ball at 3 seconds?
(c) When did the ball reach its maximum height?
(d) What was the maximum height of the ball?
(e) What is the domain and the range of the variables in the function?
2. A flare is fired into the air and its height h(t) above the ground, in meters,
after t seconds, is modeled by the function h(t) = –2t2 + 20t + 32.
(a) What is the initial height of the rocket?
(b) What was the height of the ball at 4 seconds?
(c) When did the ball reach its maximum height?
(d) What was the maximum height of the ball?
(e) What is the domain and the range of the variables in the function?
Unit 2 Chapter 6 – Quadratic Functions 38
3. A barn which contains different livestock will use 240 m of fencing to
construct three equal rectangular regions. There is no fencing along the
side of the barn so livestock can move in and out of the barn. The
quadratic function A(x) = –4x2 + 240x models the area of the pen where
A(x) represents the maximum area and x represents the width.
(a) Determine the maximum area of the pen.
(b) State the domain and range.
4. A farmer is constructing a rectangular fence in an open field to contain
cows. There is 120 m of fencing. The quadratic function A(x) = –4x2 + 240x
models the area of the pen where A(x) represents the maximum area and x
represents the width.
(a) Determine the maximum area of the pen.
(b) State the domain and range.
BARN
x x x x
240 – 4x
x x
60 – x
60 – x
Unit 2 Chapter 6 – Quadratic Functions 39
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5
-4
-3
-2
-1
1
2
3
4
5
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5
-4
-3
-2
-1
1
2
3
4
5
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5
-4
-3
-2
-1
1
2
3
4
5
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5
-4
-3
-2
-1
1
2
3
4
5
x-4 -2 2 4
y
-10
-8
-6
-4
-2
2
4
6
8
10
QUADRATIC FUNCTIONS INCLASS ASSIGNMENT REVIEW
1. Which of the following graphs represent a function?
(A) (B)
(C) (D)
2. Which of the following represent a quadratic function?
(A) y = 2x – 3 (B) y = 2x2 – 3 (C) y = –2x(x – 3) (D) y = –2x
2(x – 3) (E) y = x
2 + x + 1
3. Determine the following information from the given graph.
(a)
Direction of Opening:
Equation of Axis of Symmetry:
Vertex:
Maximum or Minimum Value:
x–intercepts: y–intercept: ____
Domain: Range:
Unit 2 Chapter 6 – Quadratic Functions 40
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5
-4
-3
-2
-1
1
2
3
4
5
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
(b)
Direction of Opening:
Equation of Axis of Symmetry:
Vertex:
Maximum or Minimum Value:
x–intercepts: y–intercept: ____
Domain: Range:
4. Determine the following information for each quadratic function given in standard form.
(a) y = 3x2 + 18x + 28 (b) y = –2x
2 – 16x – 24
(i) Direction of Opening (ii) Equation of Axis of Symmetry (iii) Vertex
(iv) Maximum or Minimum Value (v) Number of x–intercepts (vi) y–intercept
(vii) Domain (viii) Range
5(a) Given the function y = 2x2 + 8x – 1 determine the following and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
Number of x–intercepts:
Y–intercept:
Domain:
Range:
Unit 2 Chapter 6 – Quadratic Functions 41
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
(b) Given the function y = –3x2 + 6x + 5 determine the following and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
Number of x–intercepts:
Y–intercept:
Domain:
Range:
6. Determine the equation of the axis of symmetry for the parabola that:
(a) passes through the x–intercepts ( – 8 , 0 ) and ( 2 , 0 ).
(b) passes through the x–intercepts ( – 3 , 0 ) and ( 5 , 0 ).
7. Determine the direction of opening, the x–intercepts, and the y–intercept for each quadratic function
given in factored form.
(a) y = –3( x – 2 )( x – 6 ) (b) y = 4( x + 1 )( x – 3 )
8. Change the following quadratic functions into standard form.
(a) y = 3( x + 2 )( x + 4 ) (b) y = –2( x + 3 )( x – 1 )
9(a) Given the function y = 2x( x – 4 ) determine the following and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
X–intercepts:
Y–intercept:
Domain:
Range:
Unit 2 Chapter 6 – Quadratic Functions 42
t1 2 3 4 5 6
h(t)
2
4
6
8
10
12
14
16
18
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
x-6 -5 -4 -3 -2 -1 1 2 3
y
-4
-2
2
4
6
8
10
12
14
16
18
x-2 -1 1 2 3 4
y
-12
-10
-8
-6
-4
-2
2
4
(b) Given the function y = –2( x + 1 )( x – 3 ) determine the following and sketch the graph.
Equation of Axis of Symmetry: _______
Vertex: _______
Maximum or Minimum Value: ______
X–intercepts:
Y–intercept:
Domain:
Range:
10. Determine the quadratic function, of the parabolas graphed below, in factored form.
(a) (b)
11. A ball kicked into the air is represented by the function h(t) = –2t2 + 12t, where height, h, is given
in meters and time, t, is given in seconds. The path of the ball can be seen in the graph below.
(a) What is the height of the ball at 2 seconds?
(b) What is the maximum height of the ball?
(c) When does the ball reach its maximum height?
(d) How long is the ball in the air?
(e) What is the domain and the range?
Unit 2 Chapter 6 – Quadratic Functions 43
12. The trajectory of a toy rocket is represented by the function h(t) = –4t2 + 16t, where h is height
in meters and t is time in seconds.
(a) What is the initial height of the toy rocket before it takes flight?
(b) What is the height of the rocket after 3 seconds?
(c) At what time does the rocket reach its maximum height?
(d) What is the maximum height reached by the rocket
(e) What is the domain and the range of the variables in the given quadratic function?
13. A flare is fired into the air and its height h(t) above the ground, in meters, after t seconds is
modeled by the function h(t) = –2t2 + 20t + 36.
(a) What is the initial height of the flare?
(b) What was the height of the flare at 4 seconds?
(c) When did the flare reach its maximum height?
(d) What was the maximum height of the flare?
(e) What is the range of the given quadratic function?
14. A rectangular play enclosure for some dogs is to be made with 100 m of fencing the kennel
as one side of the enclosure as shown. The quadratic function that models the area of the
enclosure is represented by the function A(x) = –2x2 + 100x where A(x) represents the area
enclosed and x represents the width in meters.
(a) Determine the maximum area.
(b) What is the length and the width of the rectangular play enclosure?
(c) State the domain and the range of the variables in the quadratic function.
Kennel
Play Enclo
sure
x x
100 – 2x
Unit 2 Chapter 6 – Quadratic Functions 44
15. A rectangular region, placed against the wall of a house, is divided into three regions of equal area
using a total of 160 m of fencing as shown. The quadratic function that models the area of the
enclosure is represented by function A(x) = –4x2 + 160x where A(x) represents the area enclosed
and x represents the width in meters.
(a) Determine the maximum area of the combined three regions.
(b) What is the length and the width of the area enclosed by the fencing?
(c) State the domain and the range of the variables in the quadratic function.
ANSWERS
1. A and B 2. B, C, and E
3(a) Direction of Opening: Upwards Axis of Symmetry: x = 1 Vertex: ( 1 , – 9 )
Minimum Value of – 9 x–intercepts: x = – 2 , 4 y–intercept: y = – 8 , ( 0 , – 8 )
Domain: {x| xʀ} Range: {y| y ≥ – 9 ; yʀ}
(b) Direction of Opening: Downwards Axis of Symmetry: x = – 1 Vertex: ( – 1 , 4 )
Maximum Value of 4 x–intercepts: x = – 3 , 1 y–intercept: y = 3 , ( 0 , 3 )
Domain: {x| xʀ} Range: {y| y ≤ 4 ; yʀ}
4(a) Direction of Opening: Upwards Axis of Symmetry: x = – 3 Vertex: ( – 3 , 1 )
Minimum Value of 1 Number of x–intercepts: 0 y–intercept: y = 28 , ( 0 , 28 )
Domain: {x| xʀ} Range: {y| y ≥ 1 ; yʀ}
fencingWall of House
x x x x
160 – 4x
Unit 2 Chapter 6 – Quadratic Functions 45
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
(b) Direction of Opening: Downwards Axis of Symmetry: x = – 4 Vertex: ( – 4 , 8 )
Maximum Value of 8 Number of x–intercepts: 2 y–intercept: y = 3
Domain: {x| xʀ} Range: {y| y ≤ 8 ; yʀ}
5(a) Equation of Axis of Symmetry: x = – 2
Vertex: ( – 2 , – 9 )
Minimum Value of – 9
Number of x–intercepts: 2
Y–intercept: y = – 1
Domain: xʀ
Range: y ≥ – 9
(b). Equation of Axis of Symmetry: x = 1
Vertex: ( 1 , 8 )
Maximum Value of 8
Number of x–intercepts: 2
Y–intercept: y = 5
Domain: xʀ
Range: y ≤ – 9
6(a) x = – 3 (b) x = 1
7(a) downwards , x = 2 , 6 , y = – 36 (b) upwards , x = – 1 , 3 , y = – 12
8(a) y = 3x2 + 18x + 24 (b) y = –2x
2 – 4x + 6
Unit 2 Chapter 6 – Quadratic Functions 46
x- 2 2 4 6
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
x- 2 2 4
y
- 10
- 8
- 6
- 4
- 2
2
4
6
8
10
9(a) Equation of Axis of Symmetry: x = 2
Vertex: ( 2 , – 8 )
Minimum Value of – 8
X–intercepts: x = 0 , x = 4
Y–intercept: y = 0
Domain: xʀ
Range: y ≥ – 8
(b) Equation of Axis of Symmetry: x = 1
Vertex: ( 1 , 8 )
Maximum Value of 8
X–intercepts: x = – 1 , 3
Y–intercept: y = 6
Domain: xʀ
Range: y ≤ 8
10(a) y = –2( x + 5 )( x – 1 ) (b) y = 3( x + 1 )( x – 3 )
11(a) 16 m (b) 18 m (c) 3 seconds (d) 6 seconds (e) Domain: 0 ≤ t ≤ 6 Range: 0 ≤ h ≤ 18
12(a) 0 m (b) 12 m (c) 2 seconds (d) 16 m (e) Domain: 0 ≤ t ≤ 4 Range: 0 ≤ h ≤ 16
13(a) 36 m (b) 84 m (c) 5 seconds (d) 86 m (e) Range: 0 ≤ h ≤ 86
14(a) Maximum Area = 1250 m2 (b) Length = 50 m , Width = 25 m
(c) Domain: 0 ≤ x ≤ 50 Range: 0 ≤ A ≤ 1250
15(a) Maximum Area = 1600 m2 (b) Length = 80 m , Width = 20 m
(c) Domain: 0 ≤ x ≤ 40 Range: 0 ≤ A ≤ 1600
Unit 2 Chapter 6 – Quadratic Functions 47
6.4 – Vertex Form of a Quadratic Function
(I) Investigating the form y = a(x – p)2 + q
REMEMBER: The base graph of y = x2.
x y = x2
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 2
- 1
1
2
3
4
5
Vertex:______________
Goals:
Investigating the form y = a(x – p)2 + q
Sketching the graph of a quadratic function in Vertex Form
Determining the Equation of a Parabola from a Graph
Determining the Equation of a Parabola from a Verbal Description
Unit 2 Chapter 6 – Quadratic Functions 48
Sketch each quadratic function using https://www.desmos.com/ , state the
coordinates of the vertex and sketch the axis of symmetry.
(a) y = x2 + 1 (b) y = x
2 – 2
Example: Without the aid of a graph, determine the coordinates of the
vertex for:
(i) y = x2 – 3 (ii) y = x
2 + 5
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 2
- 1
1
2
3
4
5
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 2
- 1
1
2
3
4
5
Vertex:______________ Vertex:______________
Based on your answers to the vertex of the two graphs above, how did the base
graph of y = x2 physically shift based on the value of q in y = x2 + q?
Vertex:______________ Vertex:______________
Unit 2 Chapter 6 – Quadratic Functions 49
Sketch each quadratic function using https://www.desmos.com/ , state the
coordinates of the vertex and sketch the axis of symmetry. .
(c) y = (x + 3)2 (d) y = (x – 1)
2
Example: Without the aid of a graph, determine the coordinates of the vertex for:
(i) y = (x + 6)2 (ii) y = (x – 5)
2
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 2
- 1
1
2
3
4
5
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 2
- 1
1
2
3
4
5
Vertex:______________ Vertex:______________
Based on your answers to the vertex of the two graphs above, how did the base
graph of y = x2 physically shift based on the value of p in y = (x – p)2?
Vertex:______________ Vertex:______________
Unit 2 Chapter 6 – Quadratic Functions 50
Sketch each quadratic function using https://www.desmos.com/ , state the
coordinates of the vertex and sketch the axis of symmetry.
(e) y = (x + 3)2 + 1 (f) y = –3(x – 1)
2 – 2
Example: Without the aid of a graph, determine the coordinates of the
vertex for:
(i) y = (x + 7)2 – 2 (ii) y = 4(x – 8)
2 + 9
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 2
- 1
1
2
3
4
5
x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5
y
- 6
- 5
- 4
- 3
- 2
- 1
1
2
Vertex:______________ Vertex:______________
Based on your answers to the vertex of the two graphs above, how did the base
graph of y = x2 physically shift based on the value of p and q in y = (x – p)2 + q?
Vertex:______________ Vertex:______________
Unit 2 Chapter 6 – Quadratic Functions 51
(II) Sketching the graph of a quadratic function in vertex form.
Example:
Given the function
(a) state the direction
(b) state the coordinates of the vertex
(c) state the equation of axis of symmetry
(d) determine the y – intercept
(e) sketch the graph
(f) state the domain and range
Domain:_______________
Range:________________
A quadratic function is in vertex form when it is written in the form
y = a(x – p)2 + q
where • a indicates _______________
• coordinates of the vertex ______________
• equation of axis of symmetry ___________
x- 3 - 2 - 1 1 2 3 4 5 6 7 8
y
- 6
- 5
- 4
- 3
- 2
- 1
1
2
3
4
Unit 2 Chapter 6 – Quadratic Functions 52
(III) Determining the equation of a parabola from a graph.
Example:
Determine the equation of the quadratic function in vertex form.
(IV) Determining the equation of a parabola from a verbal description.
Example 1:
A parabola has vertex at (2, –6) and passes through the point (4, 8),
determine the function. State the range of the function.
x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2
y
- 6
- 5
- 4
- 3
- 2
- 1
1
2
Unit 2 Chapter 6 – Quadratic Functions 53
Example 2:
A parabola intercepts the x – axis at –4 and 6 and has a maximum value
of 5. Determine the function that models the parabola and state the range.
P.363 – 367 #1b, c, e #4 #5 #8 #11 #12 #14
Unit 2 Chapter 6 – Quadratic Functions 54
DETERMINING QUADRATIC FUNCTIONS IN VERTEX FORM
EXAMPLE 1
(a) Determine the quadratic function, in vertex form, for the given graph.
(b) Change the quadratic function from part (a) into standard form.
x- 4 - 2 2 4 6
y
- 4
- 2
2
4
6
8
10
Unit 2 Chapter 6 – Quadratic Functions 55
EXAMPLE 2
(a) Determine the quadratic function, in vertex form, for the given graph.
(b) Change the quadratic function from part (a) into standard form.
x- 6 - 5 - 4 - 3 - 2 - 1 1 2
y
- 4
- 2
2
4
6
8
10
Unit 2 Chapter 6 – Quadratic Functions 56
PRACTICE QUESTIONS
1(a) Determine the quadratic function, in vertex form, for the given graph.
(b) Change the quadratic function from part (a) into standard form.
x- 2 2 4 6 8
y
- 10
- 8
- 6
- 4
- 2
2
4
Unit 2 Chapter 6 – Quadratic Functions 57
2(a) Determine the quadratic function, in vertex form, for the given graph.
(b) Change the quadratic function from part (a) into standard form.
x- 6 - 4 - 2 2 4
y
- 4
- 2
2
4
6
8
10
12
Unit 2 Chapter 6 – Quadratic Functions 58
6.4 – Vertex Form of a Quadratic Function (Part II)
(I) Solving a problem that models a quadratic function.
Example 1
A basketball player taking a free throw
releases the ball at a height of 8 feet while
standing on the free throw line. At 7 feet
from the free throw line the ball attains a
maximum height of 13 ft.
(a) Determine the quadratic function that models the path of the
basketball.
(b) Determine the height of the ball when it is 3 feet from the free throw
line.
(c) Determine the domain and range.
Goals:
Solving a Problem that Models a Quadratic Function
Predicting the Number of Zeros of a Quadratic Function
Unit 2 Chapter 6 – Quadratic Functions 59
Example 2
Suppose a parabolic archway has a width of 280 cm
and a height of 216 cm at its highest point above the
floor.
(a) Write a quadratic function in vertex form that models
the shape of this archway.
(b) Determine the height of the archway that is 50 cm from its outer edge.
Unit 2 Chapter 6 – Quadratic Functions 60
(II) Predicting the number of zeros of a quadratic function.
For each quadratic function:
•state the direction •the vertex
•sketch the graph •state the number of x – intercepts
(a) y = x2 – 4 (b) y = (x – 4)
2
Direction:_____ Direction:_____
Vertex:_______ Vertex:_______
Number of x – intercepts:____ Number of x – intercepts:____
(c) y = (x – 4)2 + 4
Direction:_____
Vertex:_______
Number of x – intercepts:____
x
y
x
y
x
y
Unit 2 Chapter 6 – Quadratic Functions 61
Summary:
Example:
Predict the number of x – intercepts (or zeros) for:
(i) y = –2x2 + 4 (ii)
(iii) g(x) = –(x + 2)2
Note: A parabola may have zero, one or two x – intercepts depending on:
(i) direction
and (ii) vertex location
P.363 – 367 #2 #15 #17 #18 #19
Unit 2 Chapter 6 – Quadratic Functions 62
6.5 – Solving Problems Using Quadratic Function Models
(I) Applying quadratic models in problem solving.
Example:
A quarterback throws the ball from an initial height of 6 feet. It is caught by
the receiver 50 feet away, at a height of 6 feet. The ball reaches a maximum
height of 20 feet during its flight. Determine the quadratic function which
models this situation and state the domain and range.
Remember: Forms of a Quadratic Function
(i) Standard Form y = ax2 + bx + c
(ii) Factored Form y = a(x – r)(x – s)
(iii) Vertex Form y = a(x – p)2 + q
•Sketch a picture
Based on the information given in the problem,
select the best form for modeling the problem.
•If the vertex is given select Vertex Form
•If x-intercepts are given select Factored From
•Substitute the given information into the
quadratic form that was selected and
determine the value of ‘a’.
•Once the value of ‘a’ is determined, write
the quadratic form that models the
problem.
Goals:
Applying Quadratic Models in Problem Solving
Solving a Max/Min problem with a Quadratic Function in Standard Form
Representing a Situation with a Quadratic Model
Unit 2 Chapter 6 – Quadratic Functions 63
(II) Solving a maximum/minimum problem with a quadratic function in
standard form.
Example:
A boat in distress fires off a flare. The height of the flare, h, in metres
above the water, t seconds after shooting, is modeled by the function
h(t) = –4.9t2 + 29.4t + 3. Algebraically determine the maximum height
attained by the flare.
(III) Representing a situation with a quadratic model.
(A) Revenue Problems
Example 1
A travel agency offers a group rate of $2400 per person for a week in
London if 16 people sign up for the tour. For each additional person who
signs up, the price per person is reduced by $100. How many people, in
total, must sign up for the tour in order for the travel agency to maximize
their revenue? Determine the maximum revenue.
Formula for revenue:
Revenue = (number sold)(cost)
Unit 2 Chapter 6 – Quadratic Functions 64
Revenue Problems
Example 2
Global Gym charges its adult members $50 monthly for a membership.
The club has 600 adult members. Global Gym estimates that for each $5
increment in the monthly fee, it will lose 50 members.
(a) Determine the function that models Global Gym’s revenue.
(b) Determine the maximum revenue generated.
(c) Determine the monthly fee that will produce the greatest revenue.
Formula for revenue:
Revenue = (number sold)(cost)
Unit 2 Chapter 6 – Quadratic Functions 65
Example 3
An orange grower has 400 crates of oranges ready for market and will
have 20 more crates each day that shipment is delayed. The present price
is $60 per crate however, for each day shipment is delayed, the price per
crate decreases by $2.
(a) Determine the revenue function that models this function.
(b) Determine the maximum revenue that can be generated.
(c) Determine price per crate that will produce the greatest revenue.
Unit 2 Chapter 6 – Quadratic Functions 66
6.5 – Solving Area Problems Using Quadratic Function Models
(Part II)
(I) Open field example:
Example 1
A farmer is constructing a rectangular fence in an open field to contain
cows. There is 120 m of fencing. Write the quadratic function that models
the rectangular region, and use it to determine the maximum area of the
enclosed region.
Formula for Area of a rectangle:
Area = (width)(length)
Width =
Length =
Step I
•which variable represents width ______
Step II
•develop the expression for length as a
function of width.
•NOTE: when constructing a fence in an
open field, ________ of the fencing is used
when the first width and length sides are
constructed.
Step III
•Using the width and length expressions,
develop the area function in standard form.
Unit 2 Chapter 6 – Quadratic Functions 67
Example 2
You have 600 meters of fencing and a large field. You want to make a
rectangular enclosure split into two equal lots. Write the quadratic
function that models the rectangular region Use the function to determine the
dimensions would yield an enclosure with the largest area?
Length =
When constructing a fence in an open field,
________ of the fencing is used when half of
the width and length sides are constructed.
Width =
Unit 2 Chapter 6 – Quadratic Functions 68
(II) Using a physical structure as one side
Example 1
A Heavy Equipment Operator has 200 m of fencing to construct a
rectangular storage area using a Warehouse as one side. Write the quadratic
function that models the rectangular region, and use it to determine the
maximum area of the enclosed region.
Warehouse
Length =
•The expression for length is developed on
the basis of what fencing is left over after
both width lines have been constructed.
Width = Width =
Unit 2 Chapter 6 – Quadratic Functions 69
Example 2
A rectangular region, placed against the wall of a house, is divided into
two regions of equal area using a total of 150 m of fencing as shown.
(a) Develop a quadratic function that models the area of the pen.
(b) Determine the maximum area of the pen.
(c) State the domain and range.
fencingWall of House
Unit 2 Chapter 6 – Quadratic Functions 70
Revenue Practice Problems
1. A dinner theatre show that sold 200 tickets currently cost $40 per ticket.
Proposed decreases in ticket prices reveal that for each $2 decrease,
20 more people will attend. Write a quadratic function to model
the theatre’s revenue and use it to determine the ticket price that will
maximize profit.
2. An Airline company sells 600 tickets per flight at a cost of $100 per ticket.
Proposed increases in ticket prices reveal that for each $5 increase, 20 less
people will purchase tickets. Write a quadratic function to model the
Airline’s revenue per flight and use it to determine the maximum revenue
that can be generated per flight. What ticket price should the airline charge
to maximize revenue?
3. A dinner theatre show which sells out each night with 400 tickets currently
cost $10 per ticket. Proposed increases in ticket prices reveal that for each
$2 increase, 20 less people will attend. Write a quadratic function to model
the theatre’s revenue and use it to determine the ticket price that will
maximize profit.
4. An Airline company sells 500 tickets per flight at a cost of $100 per ticket.
Proposed increases in ticket prices reveal that for each $5 increase, 20 less
people will purchase tickets. Write a quadratic function to model the
Airline’s revenue per flight and use it to determine the maximum revenue
that can be generated per flight. What ticket price should the airline charge
to maximize revenue?
Unit 2 Chapter 6 – Quadratic Functions 71
Area Practice Problems
5. A rectangular play enclosure for some dogs is to be made with 40 m of
fencing using the kennel as one side of the enclosure as shown.
(a) Develop a quadratic function that
models the area of the pen.
(b) Determine the maximum area.
(c) State the domain and range of the variables in the function.
6. A barn which contains different livestock will use 240 m of fencing to
construct three equal rectangular regions. There is no fencing along the
side of the barn so livestock can move in and out of the barn.
(a) Develop a quadratic function that
models the area of the pen.
(b) Determine the maximum area of
the pen and state the dimensions.
Kennel
Play Enclo
sure
BARN
Unit 2 Chapter 6 – Quadratic Functions 72
x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2
y
- 6
- 5
- 4
- 3
- 2
- 1
1
2
7. A lifeguard marks off a rectangular swimming area at a beach with 200 m
of rope using the beach as one side. Determine the maximum area and the
dimensions of the swimming area?
8. A farmer is going to construct a rectangular fence in an open field using
400m of fencing. Develop an appropriate quadratic function and use it to
determine the maximum enclosed area and the dimensions of the
rectangular region.
9. A rectangular storage area for heavy equipment is to be constructed using
148 m of fencing and a building as one side. Set up an appropriate equation
and use it to determine the dimensions required to maximize the area
enclosed.
Answers: 1. R = -40x
2 + 400x + 8000 Maximum Profit = $ 9000 New Ticket Price = $ 30
2. R = -100x2 + 1000x + 60 000 Maximum Profit = $ 62 500 New Ticket Price = $ 125
3. R = -40x2 + 600x + 4000 New Ticket Price = $ 25
4. R = -100x2 + 500x + 50 000 Maximum Profit = $ 50 625 New Ticket Price = $ 112.50
5(a) A = -2x2 + 40x (b) 200 m
2 (c) domain 0 < x < 20 range
6.(a) A = -4x2 + 240x (b) 3600 m
2 30m x 120 m
7. 5000 m2 50m x 100 m 8. A = -x
2 + 200x 10 000 m
2 100m x 100 m
9. 37m x 74 m
Quiz Review
Vertex Form/Maximum – Minimum Word Problems
1. Which quadratic function has a vertex at ( 0 , – 3 ) ? 1.
(A) f(x) = ( x – 3 )2 (B) f(x) = ( x + 3 )2 (C) f(x) = x2 – 3 (D) f(x) = x2 + 3
2. Which function in Vertex Form represents the graph? 2.
(A) y = ( x + 3 )2 – 4
(B) y = ( x – 3 )2 – 4
(C) y = –( x + 3 )2 – 4
(D) y = –( x – 3 )2 – 4
Unit 2 Chapter 6 – Quadratic Functions 73
3. Which represents the number of x – intercepts for y = 2( x + 5 )2 +1? 3.
(A) 3 (B) 2 (C) 1 (D) 0
4. Which function opens down and has vertex at ( – 4 , 1)? 4.
(A) y = – ( x – 4 )2 + 1 (B) y = ( x + 4 )
2 + 1
(C) y = – ( x + 4 )2 + 1 (D) y = ( x – 4 )
2 + 1
5. Which quadratic function has an axis of symmetry at x = 2? 5.
(A) y = x2 + 2 (B) y = x
2 – 2
(C) y = ( x + 2 )2 (D) y = ( x – 2 )
2
6. What is the y–intercept for the graph of y = 2( x – 3 )2 + 1? 6.
(A) ( 0 , 19 ) (B) ( 0 , – 17 ) (C) ( 0 , 18 ) (D) ( 0 , – 18 )
7. A parabola has a vertex at ( 3 , – 4 ) and passes through the 7.
point ( 1 , 8 ). What is the quadratic function for this parabola?
(A) y = –3( x – 3 )2 – 4 (B) y = 3( x – 3 )2 – 4 (C) y = – 3( x + 3 )
2 + 4 (D) y = 3( x + 3 )
2 + 4
8. A parabola has x–intercepts at ( – 8 , 0 ) and ( 4 , 0 ) and a maximum 8.
value of 12. What is a possible quadratic function for this parabola?
(A) y = a( x + 2 )2 + 12 (B) y = a( x – 2 )
2 + 12
(C) y = a( x + 6 )2 + 12 (D) y = a( x – 6 )
2 + 12
9. A dog owner has 40 m of fencing to construct a rectangular dog pen 9.
in an open backyard. Which function represents the maximum area
of the dog pen?
(A) A = –w2 + 40w (B) A = –2w
2 + 40w
(C) A = –w2 + 20w (D) A = –2w
2 + 20w
Unit 2 Chapter 6 – Quadratic Functions 74
10. A soccer ball is kicked and follows a parabolic path described by the
function h(t) = –5t2 + 20t + 0.2, where t is the time in seconds after the
ball is kicked and h(t) is the height of the ball above ground, in meters.
(a) What is the initial height of the ball? (b) What is the height at 1 second?
(c) At what time does the ball attain the maximum height?
(d) What is the maximum height of the ball?
(e) What is the range for the variable in this situation?
11. Given the function y = 2
1( x – 4 )
2 + 3 determine the following and sketch the graph.
Direction of Opening:
Vertex:
Equation of axis of symmetry:
# of x–intercepts:
y–intercept:
Domain:
Range:
x- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10
y
- 11- 10
- 9- 8- 7- 6- 5- 4- 3- 2- 1
123456789
10
Unit 2 Chapter 6 – Quadratic Functions 75
12. Determine the function, in vertex form, represented by the given graph.
13. A quarterback throws the football from an initial height of 6 feet. It is caught
by the receiver 24 feet away, at a height of 6 feet. If the football reaches a maximum
height of 42 feet, determine the quadratic function that models the path of the
football. State the domain and range for the variables in this situation.
14. A farmer is going to construct a rectangular pen in an open field using 80 m
of fencing. Develop an appropriate quadratic function and use it to
determine the maximum enclosed area and the dimensions of the pen.
State the domain and range of the variables in this situation.
x- 2 - 1 1 2 3 4 5
y
- 5
- 4
- 3
- 2
- 1
1
2
3
4
5
6
7
8
Unit 2 Chapter 6 – Quadratic Functions 76
15. A rectangular region, placed against the wall of a house, is divided into
three regions of equal area using a total of 120 m of fencing as shown.
(a) Develop a quadratic function that models the area of the rectangular region. (b) Determine the maximum area of the rectangular region. (c) Give the length and width of the rectangular region. (d) State the domain and the range for the variables in this situation.
16. Last year, QE charged a $10 session fee for photos and 400 sessions were
booked. This year, the student council estimates that for every $1 increase
in price, they expect to have 20 fewer sessions booked.
(a) Write a quadratic function to model the maximum revenue for this situation.
(b) Determine the maximum revenue.
(c) What session fee will give the maximum revenue?
17. Global Gym charges its adult members $50 monthly for a membership.
The club has 800 adult members. Global Gym estimates that for each $5
increment in the monthly fee, it will lose 50 members.
(a) Determine the function that models Global Gym’s revenue.
(b) Determine the maximum revenue generated.
(c) Determine the monthly fee that will produce the greatest revenue.
fencingWall of House
x x x x
L =
Unit 2 Chapter 6 – Quadratic Functions 77
ANSWERS
1. C 2. A 3. D 4. C 5. D 6. A 7. B 8. A 9. C
10(a) 0.2 m (b) 15.2 m (c) 2 sec (d) 20.2 m (e) 0 ≤ h ≤ 20.2
11. Direction: down
Vertex: ( 4 , 3 )
Axis of symmetry: x = 4
# of x–intercepts: 2
y–intercept: ( 0 , – 5 )
Domain: xʀ
Range: y ≤ 3
12. y = 2( x – 2 )2 – 3
13(a) h(x) = 2
1( x – 12 )
2 + 42 (b) Domain: 0 ≤ h ≤ 24 Range: 0 ≤ h ≤ 42
14. Maximum Area = 400 m2 Dimensions: L = 20 m , W = 20 m
Domain: 0 ≤ x ≤ 40 Range: 0 ≤ A ≤ 400
15(a) A(x) = – 4x2 + 120x (b) Maximum Area = 900 m
2 (c) L = 60 m , W = 15 m
(d) Domain: 0 ≤ x ≤ 30 Range: 0 ≤ A ≤ 900
16(a) R = –20x2 + 200x + 4000 (b) Maximum Revenue = $ 4500 (c) $ 15
17(a) R = –250x2 + 1500x + 40000 (b) Maximum Revenue = $ 42 250 (c) $ 65
x- 2 - 1 1 2 3 4 5 6 7 8
y
- 6
- 5
- 4
- 3
- 2
- 1
1
2
3
4
5
Unit 2 Chapter 6 – Quadratic Functions 78
QUADRATIC FUNCTIONS TEST REVIEW
1. Which of the following represents a quadratic function opening downwards?
(A) y = 3x2(x – 1) (B) y = 3x(x – 1) (C) y = – 3x
2(x – 1) (D)y = – 3x(x – 1)
2. Which graph does NOT represent a function?
(A) (B) (C) (D)
3. Which is the y–intercept for the quadratic function y = x2 – 2x + 10?
(A) – 10 (B) 10 (C) – 2 (D) 2
4. Which quadratic function graphed below has a vertex at ( 2 , – 4 )?
(A) (B)
(C) (D)
Unit 2 Chapter 6 – Quadratic Functions 79
5. What is the axis of symmetry for the quadratic function y = –2x2 – 8x – 5?
(A) x = 2 (B) x = – 2 (C) x = 4 (D) x = – 4
6. What is the domain and range of the quadratic function graphed?
(A) Domain: {x| – 1 ≤ x ≤ 3 ; xR} Range: {x| y ≥ – 8 ; yR}
(B) Domain: {x| – 1 ≤ x ≤ 3 ; xR} Range: {x| y ≤ – 8 ; yR}
(C) Domain: {x| xR} Range: {x| y ≤ – 8 ; yR}
(D) Domain: {x| xR} Range: {x| y ≥ – 8 ; yR}
7. Which statement is correct for the function graphed below?
(A) There is a maximum value of 3. (B) There is a maximum value of 2.
(C) There is a minimum value of 3. (D) There is a minimum value of 2.
8. Determine the equation of the axis of symmetry for the parabola that passes through
the points ( – 6 , 0 ) and ( 4 , 0 ). (A) x = 2 (B) x = – 2 (C) x = 1 (D) x = –1
9. Which represents the quadratic function y = –2(x + 1)(x – 3) in standard form?
(A) y = –2x2 + 6 (B) y = –2x
2 + 4x – 6
(C) y = –2x2 – 4x – 6 (D) y = –2x
2 + 4x + 6
10. Which quadratic function opens downwards and has a vertex ( 0 , – 3 )?
(A)
y = ( x + 3 )2
(B) y = –( x + 3 )
2
(C) y = x
2 – 3
(D) y = –x
2 – 3
11. What is the equation of the axis of symmetry for the function y = –4( x – 2 )2 + 3?
(A) x = – 2 (B) x = 2 (C) x = 8 (D) x = – 8
Unit 2 Chapter 6 – Quadratic Functions 80
12. Which graph represents the function y = ( x + 4 )2 + 3?
(A) (B)
(C) (D)
(D)
13. Which quadratic function has a minimum value of 7?
(A) y = 2
1( x + 1 )
2 + 7 (B) y =
2
1 ( x + 1 )
2
+ 7
(C) y = 2
1( x + 1 )
2 – 7
(D) y =
2
1 ( x + 1 )
2 – 7
14. Which quadratic function has zero x–intercepts?
(A) y = 3( x + 4 )2 + 2 (B) y = 3( x – 4 )
2
– 2
(C) y = –3( x + 4 )2 + 2
(D) y = –3( x – 4 )
2 + 2
Unit 2 Chapter 6 – Quadratic Functions 81
15. Mark has 40 feet of lumber to enclose a rectangular flower garden. Which function
represents the area of the given flower garden, where x is the width of the garden?
(A) A(x) = –x2 + 40x (B) A(x) = x
2 + 40x
(C) A(x) = –x2 + 20x
(D) A(x) = x
2 + 20x
16. Determine the following information from the graph.
Equation of Axis of Symmetry:
Vertex:
Maximum or Minimum Value:
Y – intercept: X – intercepts: ______
Domain: Range:
17. Determine the quadratic function, of the parabola graphed below, in factored form.
x x
x- 2 2 4 6
y
- 18
- 16
- 14
- 12
- 10
- 8
- 6
- 4
- 2
2
x- 4 - 2 2 4 6
y
- 4
- 2
2
4
6
8
10
12
14
Unit 2 Chapter 6 – Quadratic Functions 82
t1 2 3 4 5
h(t)
2
4
6
8
10
12
14
18. A ball kicked into the air is represented by the function h(t) = –3t2 + 12t, where height, h, is given
in meters and time, t, is given in seconds. The path of the ball can be seen in the graph below.
(a) What is the height of the ball at 3 seconds?
(b) What is the maximum height of the ball?
(c) When does the ball reach its maximum height?
(d) How long is the ball in the air?
(e) What is the domain and the range?
Unit 2 Chapter 6 – Quadratic Functions 83
x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8
y
- 13- 12- 11- 10
- 9- 8- 7- 6- 5- 4- 3- 2- 1
12345678
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5
-4
-3
-2
-1
1
2
3
4
5
19. Given the function y = – 2x2 + 12x – 10 determine the following information and sketch the graph.
Equation of Axis of Symmetry: ___________
Vertex: __________
Maximum or Minimum value is __________
Number of x – intercepts:
Y–intercept:
Domain:
Range:
20. Determine the following information and sketch the graph of the given function.
y = 2
1( x – 2 )
2 – 3
Direction of Opening:
Vertex:
Equation of the Axis of Symmetry:
Maximum or Minimum Value:
Number of x–intercepts:
Y–Intercept:
Domain:
Range:
Unit 2 Chapter 6 – Quadratic Functions 84
21. Determine the quadratic function, in vertex form, for the given graph.
22. The trajectory of a rocket is represented by the function h(t) = – 4t2 + 16t + 20, where h is
height in meters and t is time in seconds.
(a) What is the initial height of the rocket before it takes flight?
(b) What is the height of the rocket after 3 seconds?
(c) At what time does the rocket reach its maximum height?
(d) What is the maximum height reached by the rocket?
23. A storage space is to be constructed using 100 m of wire mesh fencing. If the warehouse is to
be used as one side of the storage space, what dimensions will produce a maximum area?
What is the maximum area of the storage space?
WAREHOUSE
Unit 2 Chapter 6 – Quadratic Functions 85
24. A travel agency offers a group rate of $2000 per person for a week in
Italy if 10 people sign up for the tour. For each additional person who
signs up, the price per person is reduced by $100.
(a) Determine the revenue function.
(b) Determine the maximum revenue that can be generated.
(c) What will be the new price per person be to generate the maximum revenue?
ANSWERS:
1. D 2. D 3. B 4. B 5. B 6. D 7.A 8.D 9. D 10. D 11. B 12. A
13. A 14. A 15. C
16. Axis of symmetry x = 2
Vertex (2 , –16)
Min = –16
Y – int = (0, –12)
x – ints = (–2, 0) and (16, 0)
Domain x є R
Range y ≥ –16
17. y = –2(x + 2)(x – 3)
18.(a) 9 m (b) 12 m (c) 2 sec
(d) 4 sec (e) domain: 0 ≤ t ≤ 4 range: 0 ≤ h(t) ≤ 12
Unit 2 Chapter 6 – Quadratic Functions 86
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5
-4
-3
-2
-1
1
2
3
4
5
19. Axis of symmetry x = 3
Vertex (3 , 8)
Max = 8
Number of x – ints = 2
Y – int = (0, –10)
Domain x є R
Range y ≤ 8
20. Direction: up
Vertex (2 , –3)
Axis of symmetry x = 2
Min = –3
Number of x – ints = 2
Y – int = (0, –1)
Domain x є R
Range y ≥ –3
21. y = –2(x + 2)2 + 6
22.(a) 20 m (b) 32 m (c) 2 sec (d) 36 m
23. 25 m x 50 m Area = 1250 m2
24.(a) R = –100x2 + 1000x + 20000 (b) $22 500 (c) $1500
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