Math 20-1/20-2 Chapter...

Chapter 3 – Quadratic Functions

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Math 20-1/20-2

Chapter 3 Quadratic Functions

General Outcome: Develop algebraic and graphical reasoning through the study of relations.

Specific Outcomes: RF3. Analyze quadratic functions of the form and

determine the:

• vertex

• domain and range

• direction of opening

• axis of symmetry

• x - and y -intercepts.

[CN, R, T, V] [ICT: C6–4.3, C7–4.2]

RF4. Analyze quadratic functions of the form to identify characteristics of the

corresponding graph, including:

• vertex

• domain and range

• direction of opening

• axis of symmetry

• x - and y -intercepts

and to solve problems.

[CN, PS, R, T, V] [ICT: C6–4.1, C6–4.3]

Mark Assignments

3.1: 57– 162 # 1, 2ab, 4, 6, 7, 8, 9, 10, 21

3.2: Page 175 – 179 # (1-5, 7) (6, 9, 12, 14, 15, 23)

3.3: Page 108 – 111 # (WS1)(WS2)(13, 15, 16, 18, 23, 24)

Quiz 3 Date:

Chapter 3 Test Date:


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3.0 Exploring Quadratic Functions

Graph the following equations below. Draw a sketch next to each equation.

y= 3x – 5 y = 3x3 -8x2 - 2x + 5

y=2x y=3x2 - 2x + 4

y = -2(x - 5)2 + 1


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What are the characteristics of an equation that results in a quadratic equation?

Terminology

Degree: The largest power in an equation.

Ex: y = 3x2 - x + 1

Ex: y = 2x - 5

Ex: y = -x3 - 2x2 + 3x - 1


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3.1 – Quadratic Functions in Vertex Form

Recall: Function – a relation between two variables, x and y, where for every

input value x, there is a single output value y.

Quadratic Function – a function that has a defining equation that can be written

as 𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 or 𝒇(𝒙) = 𝒂(𝒙 − 𝒑)𝟐 + 𝒒 where a, b, and c or p and q

are constants and a ≠ 0.

This is a polynomial with the degree being __________

The graph of a quadratic function is a _____________________, and

will open up or down

Domain – the set of all _______________ (valid inputs) represented by a

graph/equation of a relation or function

Range – the set of all _______________ (valid outputs) represented by a

graph/equation of a relation or function

Example. What is the domain and range for each graph?


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Determine the Domain and Range For Each of the Following Quadratic Functions:

PROPERTIES OF A QUADRATIC FUNCTION:

VERTEX –

ex:

MAXIMUM/MINIMUM VALUE – the

highest/lowest point (y-value) of a function. For a

quadratic function, this is the y-coordinate of the

vertex.

AXIS OF SYMMETRY – a line through the vertex that

divides the graph of a quadratic function into two symmetrical

halves.

X – INTERCEPTS – the location(s) where a graph crosses the

x-axis. Also called the roots, zeros or solutions.


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Vertex: ______________

Max/Min Value:

Axis of Symmetry:

X-Intercepts:

Y- Intercept:

QUADRATIC FUNCTIONS IN VERTEX FORM:

Vertex Form: 𝒚 = 𝒂(𝒙 − 𝒑)𝟐 + 𝒒 or 𝒇(𝒙) = 𝒂(𝒙 − 𝒑)𝟐 + 𝒒

Vertex form tells you the location of the vertex (p, q) the shape of the parabola

and the direction of the opening.

Example 1 – Changing the “a” Value

On your calculator graph the following equations:

𝒚 = 𝒙𝟐

𝒚 = 𝟐𝒙𝟐

𝒚 = 𝟓𝒙𝟐

𝒚 = 𝒙𝟐

𝒚 =𝟏

𝟐𝒙𝟐

𝒚 =𝟏

𝟓𝒙𝟐

𝒚 = 𝒙𝟐

𝒚 = −𝒙𝟐

𝒚 = 𝟐𝒙𝟐

𝒚 = −𝟐𝒙𝟐

NOTICE:

“a” determines the orientation and shape of the parabola

“a” does not change the vertex or the axis of symmetry

If 𝒂 > 𝟏 𝒐𝒓 𝒂 < −𝟏, the graph is narrower compared to the graph of 𝒇(𝒙) = 𝒙𝟐

If −𝟏 < 𝒂 < 𝟏, the parabola is wider compared to the graph of 𝒇(𝒙) = 𝒙𝟐

If 𝒂 > 𝟎 the graph opens UPWARD

If 𝒂 < 𝟎 the graph opens DOWNWARD

Unit 2: Quadratics Chapter 3 – Quadratic Functions

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Example 2 – Changing the “p” Value


𝒚 = (𝒙)𝟐

𝒚 = (𝒙 − 𝟐)𝟐

𝒚 = (𝒙 − 𝟒)𝟐

𝒚 = 𝒙𝟐

𝒚 = (𝒙 + 𝟑)𝟐

𝒚 = (𝒙 + 𝟔)𝟐

NOTICE:

“p” translates the graph horizontally p units (∴ changing the vertex and axis of

symmetry)

If 𝒑 > 0 (positive number), the graph is translated RIGHT

If 𝒑 < 0 (negative number), the graph is translated LEFT

ex: 𝒚 = (𝒙 − (−𝟒))𝟐 means 𝒚 = (𝒙 + 𝟒)𝟐 and translates ____________

Example 3 – Changing the “q” value


𝒚 = 𝒙𝟐

𝒚 = 𝒙𝟐 + 𝟕

𝒚 = 𝒙𝟐 − 𝟒

NOTICE:

“q” translates the graph vertically q units (∴ changing the vertex and the

max/min value of the graph)

If 𝒒 > 0 (positive number), the graph is translated UP

If 𝒒 < 0 (negative number), the graph is translated DOWN


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Example 4

1. Graph the parabola (on the graph paper) and state: 𝒚 = 𝟐(𝒙 + 𝟏)𝟐 − 𝟑

a) the direction of the opening

b) the coordinates of the vertex

c) the equation of the axis of symmetry

d) the domain and range

e) the maximum or minimum value

f) X-Intercept(s)

g) Y-Intercept

2. Graph the parabola (on the graph paper) and state: 𝒚 = −𝟏

𝟒(𝒙 − 𝟒)𝟐 + 𝟏

a) the direction of the opening

b) the coordinates of the vertex

c) the equation of the axis of symmetry

d) the domain and range

e) the maximum or minimum value

f) X-Intercept(s)

g) Y-Intercept


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How many x - intercepts can a quadratic equation have?

Sketch an example of each one:

Example 5 – Determine the Number of X-Intercepts

Determine the number of x-intercepts for each quadratic function:

1. 𝑓(𝑥) = 0.8𝑥2 − 3 - Sketch the graph using the vertex and the “a”

- Determine the number of times the function

crosses the x-axis.

2. 𝑓(𝑥) = 2(𝑥 − 1)2

3. 𝑓(𝑥) = −3(𝑥 + 2)2 − 1


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Example 6

Determine a quadratic function in vertex form for each graph.

In order to find the equation of a quadratic function you MUST have ______________ and

_________________. The vertex subs into _________________ and the point subs into

__________________ to find the value of a.


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Example 7 – Using Transformations to Sketch Graphs

Describe how to obtain the graph of each function from the graph of 𝑦 = 𝑥2.

1. 𝑦 = (𝑥 + 4)2 − 2

2. 𝑦 = −4(𝑥 + 7)2 + 2


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Example 8 – Writing Quadratic Functions in Standard Form

Write the quadratic function in vertex form that has the given characteristics:

a. Vertex at (0, 4), congruent to 𝑦 = 5𝑥2

b. Vertex at (3, 0), passing through (4, -2)

c. Vertex at (1, -1), with y-intercept 3


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Example 9: Sketch the Following Scenario

A soccer ball is kicked from the ground. After 2 sec, the ball reaches its maximum height of 20

meters. It lands on the ground at 4 seconds.

a) Sketch the Graph

b) Determine the Equation

c) What would be the domain and range of this scenario?

d) What is the height of the ball after 1 second?

3.2 – Quadratic Functions in Standard Form (Part A)

Recall: What is the equation of a quadratic function in vertex form??

Explain the meaning of each part of the equation.


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QUADRATIC FUNCTIONS IN STANDARD FORM:

𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 or 𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄

Standard form tells you the y-intercept, the shape of the parabola and the direction of the

opening

Quadratic in Standard Form: 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄

Example 1 – Changing the “c” Value

𝒇(𝒙) = −𝒙𝟐 + 𝟒𝒙 + 𝒄

Using technology, graph on a Cartesian plane the functions that result from substituting the following c-

values into the function:

c = 10

c = 0

c = -5

What effect does “c” have on the graph?? __________________________________________


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Example 2 – Changing the “a” Value

𝒇(𝒙) = 𝒂𝒙𝟐 + 𝟒𝒙 + 𝟓

Using technology, graph on a Cartesian plane the functions that result from substituting the following a-


a = -4

a = -2

a = 1

a = 2

What effect does “a” have on the graph?? _________________________________________

Example 3 – Changing the “b” value

𝒇(𝒙) = −𝒙𝟐 + 𝒃𝒙 + 𝟓

Using technology, graph on a Cartesian plane the functions that result from substituting the following b-


b = 2

b = 0

b = -2

b = -4

What effect does “b” have on the graph?? __________________________________________


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Example 4 – Identifying Characteristics of a Quadratic Function in

Standard Form

From each graph of a quadratic

function, identify the following:

i. the direction of the opening

ii. the coordinates of the vertex

iii. the maximum or minimum value

iv. the equation of the axis of

symmetry

v. the domain and range

vi. X-Intercept(s)

vii. Y-Intercept


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Example 5

Determine the equation of the axis of symmetry, if the parabola contains the points (-8, 4)

and (6, 4).

Example 6 – Functions in Standard Form

Write each function in standard form. Is it a quadratic function?

a) 𝒚 = (𝟓 − 𝒙)(𝟔 − 𝒙) b) 𝒚 = 𝒙(𝒙 + 𝟑)(𝒙 − 𝟐)


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3.2 – Quadratic Functions in Standard Form (Part B)

Recall: What is the equation of a quadratic function in vertex form?? What

is the equation of a quadratic function in standard form?? How are these two

equations related??

For any quadratic function in standard form, the x-coordinate of the vertex is

given by:

Example 1: For each of the following functions, determine the vertex:

𝒚 = 𝒙𝟐 + 𝟔𝒙 + 𝟓 𝒚 = 𝟑𝒙𝟐 − 𝟏𝟐𝒙 + 𝟓


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Example 2: Graph the following quadratic function:

𝒚 = −𝒙𝟐 + 𝟐𝒙 + 𝟑

State the following then sketch the graph

i. the direction of the opening

ii. the coordinates of the vertex

iii. the maximum or minimum value

iv. the equation of the axis of symmetry

v. the domain and range

vi. X-Intercept(s)

vii. Y-Intercept


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Example 3 – Determining the Number of x-intercepts

How many x-intercepts does each function have?

a) a quadratic function with an axis of symmetry of x=0 and a maximum

value of 8.

b) a quadratic function with a vertex at (3,1), passing through the point (1, -

3).

c) a quadratic function with a range of y ≥ 1

d) a quadratic function with a y-intercept of 0 and an axis of symmetry of x

= -1.


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Example 4 – Analyzing a Quadratic Function

A frog sitting on a rock jumps into a pond. The height, h, in centimetres, of

the frog above the surface of the water as a function of time, t, in seconds, in

since it jumped can be modeled by the function ℎ(𝑡) = −490𝑡2 + 150𝑡 +

25. Where appropriate, answer the following questions to the nearest tenth.

a) Graph the function using your graphing calculator. Sketch the graph of

the function and state your window settings.

b) What is the y-intercept? What does it represent in this situation?

c) What maximum height does the frog reach? When does it reach that

height?

d) When does the frog hit the surface of the water?

e) What are the domain and range of this situation?

f) How high is the frog 0.25 s after it jumps?


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Example 5 – Write a Quadratic Function to Model a Situation

A rancher has 100 m of fencing available to build a rectangular corral.

a) Write a quadratic function in standard form to represent the area of the

corral.

b) What are the coordinates of the vertex? What does the vertex represent in

this situation?

c) Sketch the graph for the function you determined in

part a).

d) Determine the domain and range for this situation.

e) Identify any assumptions you made in modelling this situation

mathematically.


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Example 6 – Sketch the graph of a quadratic function that has the

characteristic described in each part. Label the coordinates of three points

that you know are on the curve.

a) x-intercepts at -1 and 3 and a range b) one of its x-intercepts at -5

and a vertex

of y ≥ -4 at (-3, -4)

c) axis of symmetry of x = 1, d) vertex at (2, 5) and y-intercept of 1

minimum value of 2, and passing

through (-1, 6)

3.3 – Completing the Square (Part A)

EXPLORATION:

Recall: How is vertex form and standard form related?

PERFECT SQUARE TRINOMIALS:

Trinomials that can be factored and written as a binomial squared.

Examples:

1. 𝑥2 + 6𝑥 + 9 = (𝑥 + 3)(𝑥 + 3) = (𝑥 + 3)2

2. 𝑥2 − 4𝑥 + 4 = (𝑥 − 2)(𝑥 − 2) = _____________

3. 𝑥2 − 10𝑥 + 25 = ________________________ = ______________________

4. 𝑥2 + 18𝑥 + 81 = ________________________ = ______________________

Quadratic Functions

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Example 1: For each of the following, determine the value of “c” that would

make each trinomial expression a perfect square.

1. 𝑥2 + 8𝑥 + 𝑐

2. 𝑥2 − 6𝑥 + 𝑐

3. 𝑥2 + 14𝑥 + 𝑐

4. 𝑥2 − 2𝑥 + 𝑐

How do you determine “c”?

For each of the above trinomials, what is the equivalent binomial square expression for

each in the form (𝒙 − 𝒑)𝟐?

1. ________________________________________

2. ________________________________________

3. _________________________________________

4. _________________________________________

How do you determine “p”?

Quadratic Functions

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Completing the Square is an algebraic process used to write a quadratic

polynomial in the form 𝑎(𝑥 − 𝑝)2 + 𝑞. Completing the square involves adding a

value to and subtracting a value from a quadratic polynomial so that it contains a

perfect square trinomial. This trinomial can then be factored into the square of a

binomial!

STEPS FOR COMPLETING THE SQUARE:

1. Write the quadratic function in standard form.

2. Group the first two terms (and move the constant over to the right – out

of the way).

3. Factor out the leading coefficient from the first TWO terms (if

necessary). This is only necessary if a ≠ 1.

4. Add and subtract the square of half of the coefficient of the x-term. This

will create your perfect square trinomial.

5. Group the perfect square trinomial (by REMOVING the subtracted

number).

6. Rewrite the perfect square trinomial as the square of a binomial

(FACTOR)!

7. Simplify!!!

Quadratic Functions

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EXAMPLES:

Complete the square to write each quadratic function in vertex form.

1. 𝑥2 + 2𝑥 + 5

2. −8𝑥 + 𝑥2 − 7

3. 𝑥2 − 9𝑥 − 12 (Use decimals!)

4. 𝑥2 + 5𝑥 − 2 (Use fractions!)

Quadratic Functions

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3.3 – Completing the Square (Part B)

In order to convert from standard form to vertex form if a≠1, it will be

necessary to:

______________________________________________________________.

RECALL: STEPS FOR COMPLETING THE SQUARE:

Quadratic Functions

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EXAMPLE 1:

Complete the square to write each quadratic function in vertex form.

1. 𝑦 = 2𝑥2 + 16𝑥 + 24

What is the maximum value and when does it occur?

2. 𝑦 = 9𝑥 − 2 − 4𝑥2

What is the equation for the axis of symmetry? What is the y-coordinate of the vertex?

3. −1

2𝑥2 − 3𝑥 − 8

Quadratic Functions

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EXAMPLE 2:

Convert the function 𝑦 = −5𝑥2 − 70𝑥 to vertex form. Verify that the two forms are equivalent

both algebraically and graphically.

EXAMPLE 3:

Draw the graph of 𝑦 = −2𝑥2 − 8𝑥 − 3.

Quadratic Functions

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3.3 – Completing the Square (Part C)

INTRO:Consider the following quadratic function in vertex form:

𝒚 = 𝟑(𝒙 − 𝟐)𝟐 + 𝟏𝟔

This equation shows a minimum of __________ when ____________.

This relationship will help us in problem solving situations:

1. The following equation shows a relationship between height (m) and the time (s)

elapsed: 𝒚 = −𝟓(𝒙 − 𝟖)𝟐 + 𝟒𝟎

This relationship shows a _______________ of ______ when _______________.

2. The following equation shows a relationship between the area of a rectangle

(m2) and its length (m): 𝒚 = −𝟖(𝒙 − 𝟓)𝟐 + 𝟏𝟓

This relationship shows a _______________ of ______ when _______________.

3. The following equation shows a relationship between the vertical height (m) of a

golf ball and its horizontal distance travelled (m): 𝒚 = −𝟓(𝒙 − 𝟑𝟎)𝟐 + 𝟏𝟎

This relationship shows

Quadratic Functions

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4. The following equation shows a relationship between the Cost ($) and the

number of sweaters ordered: 𝒚 = −𝟑(𝒙 − 𝟏𝟎)𝟐 + 𝟐𝟎𝟎


5. The following equation shows a relationship between Amy’s mark on a test (%)

and the number of hours spent studying: 𝒚 = −𝟒(𝒙 − 𝟏𝟓)𝟐 + 𝟐𝟓


So far when solving problems, we have used graphing to determine the vertex and therefore, the

maximum or minimum value. Determining the max/min value by completing the square can be

faster and more efficient.

EXAMPLE 1:

Two numbers have a sum of 20. Does the sum of their squares have a maximum or minimum

value? Determine this value and the two numbers.

Quadratic Functions

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You Try …

Two numbers have a difference of 18. Does their product have a maximum or minimum value?

Determine this value and the two numbers.

EXAMPLE 2:

The perimeter of a rectangle is 32 cm. Determine the dimensions that produce a maximum area.

Quadratic Functions

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You Try …

A person has 100 m of fencing to enclose a rectangular garden. What are the dimensions of the

largest possible garden?

EXAMPLE 3:

A student parking pass costs $20. At this price, 150 students will purchase passes. For every $5

increase in price, 20 fewer students will purchase passes.

a) What is the price of a parking pass that will maximize the revenue?

b) What is the maximum revenue?

Quadratic Functions

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You Try …

Every week, a take-out restaurant sells approximately 2000 chicken wraps for $1.50 each.

Through market research, the restaurant manager determines that for every $0.10 increase in

price, she will sell 100 fewer wraps.

a) What is the price of a wrap that will maximize the revenue?

b) What is the maximum revenue?

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