Unit 2 Chapter 6 Quadratic Functions - WordPress.com · Unit 2 Chapter 6 – Quadratic Functions 5...

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


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


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 = ______


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


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?

https://www.desmos.com/


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


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


(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.


(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)


(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?



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.


(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


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.


Determining Domain & Range Graphically

1. State the domain using set notation for:

(a) (b)

Set notation:________________ Set notation:________________

(c) (d)


2. State the range using set notation for:

(a) (b)


-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


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


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.


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


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:


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.


Vertex: _______



Y–intercept:

Domain:

Range:


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.


Vertex: _______



Y–intercept:

Domain:

Range:

2. Given the function y = –2x2 + 8x – 10 determine the characteristics and sketch the graph.


Vertex: _______



Y–intercept:

Domain:

Range:


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.


Vertex: _______



Y–intercept:

Domain:

Range:

4. Given the function y = x2 + 6x + 9 determine the characteristics and sketch the graph.


Vertex: _______



Y–intercept:

Domain:

Range:


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


(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 ?


(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



(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


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 _________________


(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


Example 2: Given the quadratic function y = 2x( x + 4)






(f) state the range

x-6 -5 -4 -3 -2 -1 1 2

y

-10

-8

-6

-4

-2

2


Example 3: Given the quadratic function y = –3(x – 1)(x – 1)






(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


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


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


(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


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.


(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.







(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


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.


(b) State the domain and range of the variables in the function.

fencingWall of House

x x x x

120 – 4x


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.






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.







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


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:


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)



Vertex:


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.


Vertex: _______



Y–intercept:

Domain:

Range:


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.


Vertex: _______



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.


Vertex: _______


X–intercepts:

Y–intercept:

Domain:

Range:


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.


Vertex: _______


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?


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.


(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


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}


x x x x

160 – 4x


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


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


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


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


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


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:______________




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:______________


graph of y = x2 physically shift based on the value of p in y = (x – p)2?

Vertex:______________ Vertex:______________




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:______________


graph of y = x2 physically shift based on the value of p and q in y = (x – p)2 + q?

Vertex:______________ Vertex:______________



(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



(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


(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


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


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


EXAMPLE 2

(a) Determine the quadratic function, in vertex form, for the given graph.


x- 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 4

- 2

2

4

6

8

10


PRACTICE QUESTIONS

1(a) Determine the quadratic function, in vertex form, for the given graph.


x- 2 2 4 6 8

y

- 10

- 8

- 6

- 4

- 2

2

4


2(a) Determine the quadratic function, in vertex form, for the given graph.


x- 6 - 4 - 2 2 4

y

- 4

- 2

2

4

6

8

10

12


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


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.


(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


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


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


(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)


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)


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.


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.


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 =


(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 =


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.



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?


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


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


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


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.


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


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


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.


x x x x

L =


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

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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)


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


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


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.


Vertex:


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

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- 10

- 8

- 6

- 4

- 2

2

x- 4 - 2 2 4 6

y

- 4

- 2

2

4

6

8

10

12

14


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?


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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


Vertex:

Equation of the Axis of Symmetry:



Y–Intercept:

Domain:

Range:


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


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


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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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