Unit 3B: Quadratic Functions · Algebra 1 Unit 3B: Quadratic Functions Notes 2 Day 1: Quadratic...

Algebra 1 Unit 3B: Quadratic Functions Notes

1

Unit 3B: Quadratic Functions After completion of this unit, you will be able to…

Learning Target #1: Characteristics of Quadratic Functions Identify transformations from an function or graph

Create a function to describe given transformations

Describe characteristics of a quadratic function on a graph (vertex, axis of symmetry, intercepts, zeros, intervals of

increase/decrease, extrema, positive/negative areas, and end behavior)

Learning Target #2: Different Forms of Quadratic Functions and their Graphs Graph quadratics in vertex, standard, and factored form

Convert functions between standard, factored, and vertex form

Calculate the vertex of a function

Compare equations in multiple forms

Learning Target #3: Applications of Quadratic Functions Calculate average rate of change from a graph, table, or equation.

Calculate the vertex from a word problem and use it to answer real world questions

Compare quadratic functions using multiple representations

Timeline for Unit 3B

Monday Tuesday Wednesday Thursday Friday

February 25th

Day 1 –

Transformations of

Quadratic Functions

(h and k)

26th

Day 2 –

Transformations of

Quadratic Functions

(a value)

27th

Day 3 –

Characteristics of

Quadratic Functions

28th

Day 4 –

Characteristics of

Quadratic Functions

March 1st

Quiz over

Days 1 – 4

4th

Day 5 –

Graphing in Vertex

Form

5th

Day 6 –

Graphing in Standard

Form

6th

Day 7 –

Graphing in Factored

Form

7th

Day 8 –

Writing Equations of

Parabolas

8th

Day 9 – Finding the

Vertex by

Completing the

Square

11th

Day 10 –

Comparing Different

Forms of Quadratics

12th

Quiz over

Days 5 – 10

13th

Day 11 –

Average Rate of

Change

14th

Day 12 –

Applications of the

Vertex

15th

Day 13 –

Comparing Different

Quadratic Functions

18th

Unit 3B Review

19th

Unit 3B Test


2

Day 1: Quadratic Transformations (H & K values)

The parent function of a function is the simplest form of a function. The parent function for a quadratic function

is y = x2 or f(x) = x2. Graph the parent function below.

Vertex: ______________

x x2

-3

-2

-1

0

1

2

3

Variable Summary of the Effects of the Transformations

a

Up:

Stretch:

Down: Shrink:

h

Left:

Right:

k

Up:

Down:

As you can see, the graph of a

quadratic function is very different than

the graph of a linear function.

The U-shaped graph of a quadratic

function is called a

____________________________.

The highest or lowest point on a

parabola is called the

____________________________.

One other characteristic of a quadratic

equation is that one of the terms is

always _____________________.

Vertex Form

Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


3

Exploring the K value

1. Describe how the dotted graph has

been transformed from y = x2.

2. What is the vertex? ____________

3. How does the equation of the graph

related to its vertex?










related to its vertex? y = x2 + 3

y = x2 + 1

y = x2 - 2


4

Practice: Identify the transformations and vertex from the equations below.

1. y = x2 + 5 2. y = x2 – 3 3. y = x2 + 7 4. y = x2 - 4

Practice: Describe the transformations and name the vertex. Create an equation for the graphs listed below.

Practice: Given the transformations listed below, create an equation that would represent the transformations.

1. Shifted up 8 units 2. Shifted up 20 units 3. Shifted down 5 units

The k Value

y = a(x – h)2 + k

_________________________ if ___________________

_________________________ if ___________________


5

Exploring the H value











y = (x – 3)2

y = (x + 4)2






y = (x + 1)2


6


1. y = (x – 4)2 2. y = (x + 6)2 3. y = (x – 7)2 4. y = (x + 3)2



1. Shifted right 8 units 2. Shifted left 20 units 3. Shifted left 5 units

The h Value

y = a(x – h)2 + k

_________________________ if ___________________

_________________________ if ___________________


7

Putting It All Together


1. y = (x – 2)2 + 4 2. y = (x + 3)2 - 2 3. y = (x – 9)2 – 5 4. y = (x + 5)2 + 6



1. Shifted up 4 units and left 3 units 2. Shifted right 5 units and down 2 units

3. Shifted left 8 units and down 1 unit 4. Shifted up 5 units and right 9 units


8

Day 2: Quadratic Transformations (A values)

So far, we have discussed what the H and K values do when a quadratic function is in vertex form. How do you

think the “a” coefficient will affect the graph? The “a” value affects the graph in two different ways wh ich you

will learned about in this lesson.

h = _______ h = _____________________________

k = _____________________________

Vertex: _____________

The A Value, Part 1

Vertex Form

Vertex Form

1. Describe how the dotted graph has been

transformed from y = x2.


3. How does the equation of the graph related

to its vertex?

y = 8x2 y = 3x2

1. Describe how the dotted graph has been

transformed from y = x2.


3. How does the equation of the graph related

to its vertex?


9

How do you think the number in front affects the graph?

The a Value, Part 1

_________________________ if ___________________

_________________________ if ___________________

1. Describe how the dotted graph

has been transformed from y = x2.

2. What is the vertex? ________

3. How does the equation of the

graph related to its vertex?






y = (1/4)x2

y = (1/10)x2


10

The A Value, Part 2

Practice: Describe the transformations from the given function to the transformed function.

a. f(x) = x2 f(x) = 4x 2 b. y = x2 y = ¼x2 c. f(x) 6 f(x)

d. f(x) = x2 f(x) = -x 2 f. y = x2 y = -½x2 g. f(x) -4f(x)






The a Value, Part 2

_________________________ if ___________________

_________________________ if ___________________

y = -x2


11

Putting It All Together

Practice: Given the equations below, name the vertex and describe the transformations:

a. y = -(x – 4)2 + 7 b. y = -2(x + 2)2 + 5 c. y = ½(x – 3)2 – 8

Practice: Create an equation to represents the following transformations:

a. Shifted down 4 units, right 1 unit, and reflected across the x-axis

b. Shifted up 6 units, reflected across the x-axis, and stretch by a factor of 3

c. Shifted up 2 units, left 4 units, reflected across the x-axis, and shrunk by a factor of ¾.


12

Day 3 - Characteristics of Quadratics

Domain and Range

Domain

Define:

All possible values of x

Think:

How far left to right does the

graph go?

Write:

Smallest x ≤ x ≤ Biggest x

*use < if the circles are open*

Range Define:

All possible values of y

Think:

How far down to how far up does

the graph go?

Write:

y ≤ highest y value (opens down)

y ≥ lowest y value (opens up)

Graph 1 Graph 2

Domain: Domain:

Range: Range:

Graph 3 Graph 4

Domain: Domain:

Range: Range:

Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


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Zeros and Intercepts

Y-Intercept

Define:

Point where the graph crosses the

y-axis

Think:

At what coordinate point does the

graph cross the y-axis?

Write:

(0, b)

X-Intercept Define:

Point where the graph crosses the

x-axis

Think:

At what coordinate point does the

graph cross the x-axis?

Write:

(a, 0)

Zero

Define:

Where the function (y-value)

equals 0

Think:

At what x-value does the graph

cross the x-axis?

Write:

x = ____

Graph 1 Graph 2

X-intercepts: Y-intercept: X-intercepts: Y-intercept:

Zeros: Zeros:

Graph 3 Graph 4

X-intercepts: Y-intercept: X-intercepts: Y-intercept:

Zeros: Zeros:


14

Vertex & Axis of Symmetry

Vertex

Define:

Highest or lowest point or peak of

a parabola

Think:

What is my highest or lowest point

on my graph?

Write:

Name the point (h, k)

Axis of Symmetry Define:

The vertical line that divides the

parabola into mirror images and

runs through the vertex

Think:

What imaginary, vertical line would

make the parabola symmetrical?

Write:

x = h

(x value of the vertex)

Graph 1 Graph 2

Vertex: Vertex:

Axis of Symmetry: Axis of Symmetry:

Graph 3 Graph 4

Vertex: Vertex:

Axis of Symmetry: Axis of Symmetry:


15

Extrema

Maximum Define:

Highest point or peak of a

function.

Think:

What is my highest point on my

graph?

Write:

y = k

(y-value of the vertex)

Minimum Define:

Lowest point or valley of a

function.

Think:

What is the lowest point on my

graph?

Write:

y = k

(y-value of the vertex)

Graph 1 Graph 2

Extrema: Extrema:

Min/Max Value: Min/Max Value:

Graph 3 Graph 4

Extrema: Extrema:

Min/Max Value: Min/Max Value:


16

End Behavior

End Behavior Define:

Behavior of the ends of the function (what happens to the y-values or f(x)) as x approaches positive or

negative infinity. The arrows indicate the function goes on forever so we want to know where those ends go.

Think:

As x goes to the left (negative infinity), what direction

does the left arrow go?

Write:

As x -∞, f(x) _____

Think:

As x goes to the right (positive infinity), what direction

does the right arrow go?

Write:

As x ∞, f(x) _____

Graph 1 Graph 2

As x -∞, f(x) _______. As x -∞, f(x) _______.

As x ∞, f(x) _______. As x ∞, f(x) _______.

Graph 3 Graph 4

As x -∞, f(x) _______. As x -∞, f(x) _______.

As x ∞, f(x) _______. As x ∞, f(x) _______.


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Day 4 - Characteristics of Quadratics (Cont’d)

Intervals of Increase and Decrease

Interval of Increase

Define:

The part of the graph that is

rising as you read left to right.

Think:

From left to right, is my graph

going up?

Write:

An inequality using the x-value of the vertex

Interval of Decrease Define:

The part of the graph that is

falling as you read from left

to right.

Think:

From left to right, is my graph

going down?

Write:

An inequality using the x-value of the vertex

Graph 1 Graph 2

Interval of Increase: Interval of Increase:

Interval of Decrease: Interval of Decrease:

Graph 3 Graph 4

Interval of Increase: Interval of Increase:

Interval of Decrease: Interval of Decrease:


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Positive & Negative Parts of the Graph

Positive Define:

The part of the

function that is

above the x-axis.

Think:

Which part of the

function is in the

positive region

and where?

Write:

Inequality using

the zeros value (x)

Negative Define:

The part of the

function that is

below the x-axis.

Think:

Which part of the

function is in the

negative region

and where?

Write:

Inequality using

the zero values (x)

Graph 1 Graph 2

Positive: Positive:

Negative: Negative:

Graph 3 Graph 4

Positive: Positive:

Negative: Negative:


19

Practice: Describe the characteristics of the following graphs:

Domain: ___________________ Range: ______________________

Vertex: ____________________ Axis of Sym.__________________

Y-Intercept: ________________ Zeroes: ______________________

Extrema: ___________________ Max/Min Value: _____________

Int of Inc: __________________ Int of Dec: ___________________

Positive: ____________________ Negative: ____________________

End Behavior: As x -∞, f(x) ______. As x ∞, f(x) ______

Domain: ___________________ Range: ______________________







Domain: ___________________ Range: ______________________








20

Day 5 - Graphing Quadratics in Vertex Form

a determines how the graph opens

positive a, graph opens ____________

negative a, graph opens ____________

&

(_____ , _____) is our vertex.

NOTE: Our vertex is at (h, k), NOT (-h, k).

Identifying the Vertex Practice

Find the vertex of the following:

1) y = (x – 18)2 + 9 Vertex = (_____ , _____)

2) y = 4(x + 6)2 – 7 Vertex = (_____ , _____)

3) y = (x – 2)2 -2 Vertex = (_____ , _____)

Find the vertex for each of the following quadratics and determine whether the graph opens up or down:

a) y = (x -1)2 – 2 Vertex = (_____ , _____) Graph Opens ___________ because a is ___

b) y = -3(x + 4)2 + 1 Vertex = (_____ , _____) Graph Opens ___________ because a is ___

c) y = 2x2 + 3 Vertex = (_____ , _____) Graph Opens ___________ because a is ___

d) y = -(x - 3)2 Vertex = (_____ , _____) Graph Opens ____________ because a is ___

Vertex Form of a Quadratic Function:

y= a(x - h)2 + k

Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


21

Steps for Graphing in Vertex Form

1) Find the vertex (h, k).

2) Use your vertex as the center for your table and determine two x values to the left and right of your h

value and substitute those x values back into the equation to determine the y values.

- Using practice problem number 3, let’s practice filling in our table.

y = (x – 2)2 - 2

3) Plot your points and connect them from left to right!

Graphing in Vertex Form Examples

Example 1: Graph y = (x -1)2 – 2.

Vertex = (_____ , _____)

Example 2: Graph: y = -3(x + 4)2 + 1.

Vertex = (_____ , _____)

x

y

x -1 0

2 3

y

x

y

-2

-11

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


22

Example 3: Graph y = 2x2 + 3.

Vertex = (_____ , _____)

Example 4: Graph: y = -(x - 3)2.

Vertex = (_____ , _____)

Using a Graphing Calculator to Graph Quadratics in Vertex Form

Use a graphing calculator to graph our last example problem, example 4: y= -(x - 3)2

1. Hit Y = and enter the equation into y1.

2. Hit Graph (Hit Zoom, then 6 to get back to a standard viewing window, if necessary).

3. You can also use the table on the graphing calculator to compare to your table and note the symmetry

along the vertex. Hit 2nd followed by Graph (you really want the Table feature). Scroll through the table until you

find where the y1 values stop decreasing and begin increasing, the point it switches at is our vertex.

x

y

x

y

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


23

Day 6 - Graphing Quadratics in Standard Form

Given the following equation, y = (x + 3)2 + 1, how could we go from that form to y = x2 + 6x + 10?

What about y = 3(x + 2)2 + 3 to y = 3x2 +12x +15?

This is how we arrive to the standard form of a quadratic function!

A determines how the graph open

(0, C) is the y-intercept.

Finding the Vertex in Standard Form

To graph in standard form, we need to find the vertex.

The formula for the x-coordinate is:

2

bx

a, then substitute x into equation for y.

For example, say we have y = x2 + 2x + 7, how would we find our vertex?

Standard Form of a Quadratic Function:

y= Ax2 + Bx + C

Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


24


Find the vertex for each of the following quadratics, determine whether the graph opens up or down, and find

the y intercept:

1. y = 2x2 + 8x + 2 Vertex = (_____ , _____) 2. y = -x2 + 2x + 7 Vertex = (_____ , _____)

Graph opens ___________ because a is ______. Graph opens ___________ because a is ______.

The y-intercept is (0, ). The y-intercept is (0, ).

3. y = - 4x2 + 24x Vertex = (_____ , _____) 4. y = 7x2 + 9 Vertex = (_____ , _____)

Graph opens ___________ because a is ______. Graph opens ___________ because a is ______.

The y-intercept: ___________ The y-intercept: ___________

Steps for Graphing in Standard Form

1) Find the vertex. After using the formula

2

bx

ato find our x- coordinate of our vertex, we substitute that x

back into our equation, and our solution is the y-coordinate of our vertex.

2) Use your vertex as the center for your table and determine two x values to the left and right of your x-

coordinate and substitute those x values back into the equation to determine the y values.

3) Plot your points and connect them from left to right!


25

Graphing in Standard Form Examples

Example 1: Graph y = x2 – 2x – 1.

Vertex = (_____ , _____)

Example 2: Graph: y = 3x2 – 6x.

Vertex = (_____ , _____)

Example 3: Graph y = 2x2 + 3.

Vertex = (_____ , _____)

x -1 0

2 3

y

x

y

0

9

x

y

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


26

Example 4: Graph: y = - x2 + 6x – 9.

Vertex = (_____ , _____)

Using a Graphing Calculator to find the Vertex of Quadratics in Standard Form

We already know how to graph quadratics, so let’s try and find the vertex of these equations using our

graphing calculators! Graph y = x2 + 2x - 3

1. Hit Y = and enter the equation into y1.

2. Hit 2nd followed by Trace (you really want the calc function). If your

parabola OPENS UP select 3: minimum, if your parabola OPENS DOWN

select 4: maximum.

3. (You may have to move the spider left and

right using your arrow buttons). The calculator will ask you “left bound?” hit

Enter. The calculator will then ask you “right

bound?” hit Enter. The calculator will then ask

you “guess?” hit Enter.

4. Your maximum or minimum coordinates will be displayed on the screen

and that is your vertex!

x

y

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


27

Day 7 - Graphing Quadratics in Intercept (Factored) Form

We learned in Unit 3A how to factor, but we can also graph in factored form!

a determines how the graph opens

The x - intercepts are (p, 0) and (q, 0).

Finding the Vertex in Intercept Form

Graphing in factored form is similar to graphing in standard form, but the way we find our vertex is different. We

use a special formula to find the x - coordinate of our vertex, and substitute that value in our equation to

determine the y - coordinate of our vertex. The formula is:

𝑥 = 𝑝 + 𝑞

2

For example, say we have y = (x + 7) (x + 1), how would we find our vertex?


Find the vertex for each of the following quadratics and determine the x- intercepts:

1. y = (x +1)(x + 3) 2. y = (x + 4)(x – 2)

x-intercepts: __________ & ___________ x-intercepts: __________ & ___________

Vertex = _____________ Vertex = _____________

Factored Form of a Quadratic Function:

y= a(x - p)(x – q)

Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


28

Steps for Graphing in Intercept Form

1. Find the vertex. After using the formula x = p+q

2 to find our x- coordinate of our vertex, we substitute that x

back into our equation, and our solution is the y-coordinate of our vertex.

2. Determine your two x – intercepts.

3. Plot your points and connect them from left to right!

Graphing in Factored Form Examples

Example 1: Graph y= (x + 2)(x – 2).

x – intercepts:

Vertex:

Example 2: Graph: y= - (x +1)(x -7).

x – intercepts:

Vertex:

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


29

Example 3: Graph y= 2(x -1)(x - 3).

x – intercepts:

Vertex:

Example 4: Graph: y= -(x - 3)2

x – intercepts:

Vertex:

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


30

Day 8 – Writing Equations of Parabolas from a Graph

From days 5 – 8, you learned about three different forms of quadratic functions – vertex, standard, and

factored form. Each form tells you something different about the graph.

Vertex Form Standard Form Intercept Form

(Factored Form)

y = a(x – h)2 + k

(h, k) is the vertex

y = ax2 + bx + c

c is the y-intercept

y = a(x – p)(x – q)

p and q are x-intercepts

a always determines the way the graph opens

Writing Equations of Parabolas Given a Graph For the following graphs:

A. Create an equation in both intercept and vertex form to describe the parabola. Assume there are

no stretches or shrinks with each graph.

B. Once you created both equations, convert both to standard form. Check to make sure the y-

intercepts match both the graph and the equations in standard form.

C. Put all three equations into your graphing calculator. Do you get the same graph for all three

equations?

a. Intercept Form Vertex Form

Standard Form Standard Form

Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


31

b. Intercept Form Vertex Form


c. Intercept Form Vertex Form


d. Intercept Form Vertex Form



32

Day 9 – Finding the Vertex by Completing the Square

Think About It: Earlier in this unit, you learned to find the vertex of an equation in standard form by finding the x-

value of the vertex using x = -b/2a. Today, you are going to learn how to use completing the square to find the

vertex.

Take a look at the graph and conversion to standard form. How would you go from g(x) = x2 + 8x + 11 to

g(x) = (x + 4)2 – 5?

Finding the Vertex by Completing the Square

To finding the vertex from standard form, we are only going to focus on the right side of the equation. Take a

look at the following example from above, but this time, we are going from standard to vertex.

Steps Reasoning/Justification

y = x2 + 8x + 11 Original Equation

y = (x2 + 8x) + 11

When completing the square, we only want to consider the x2 & x

terms

y = (x2 + 8x + ____) + 11 - ____

Since we are only working on one side of the equation, we want to

add and subtract whatever number allows us to “complete the

square” so the function doesn’t change in value (we are technically

adding zero).

y = (x2 + 8x + 16) + 11 - 16 When completing the square, take half of the b-value and square it.

y = (x + 4)2 + 11 - 16 We can rewrite the perfect square trinomial as a binomial square

(essentially we factored x2 + 8x + 16).

y = (x + 4)2 – 5

Vertex: (-4, -5) Combine the like terms (11 & -16).


33

Practice Finding the Vertex by Completing the Square

Find the vertex of the quadratic functions by completing the square.

a. f(x) = x2 + 6x + 11 b. y = x2 – 10x + 2

c. g(x) = x2 + 4x d. y = x2 – 5x + 4

e. y = 2x2 – 12x + 16 f. h(x) = -2x2 + 8x – 4


34

Day 10 – Comparing Different Forms of Quadratic Functions

From days 5 – 8, you learned about three different forms of quadratic functions – vertex, standard, and

factored form. Each form tells you something different about the graph.

Vertex Form Standard Form Intercept Form

(Factored Form)

y = a(x – h)2 + k

(h, k) is the vertex

y = ax2 + bx + c

c is the y-intercept

y = a(x – p)(x – q)

p and q are x-intercepts

a always determines the way the graph opens

Practice: Given a quadratic equation below, name the form the equation is in and describe the characteristics

you gain from that form. Some equations might be considered two different forms.

a. y = -3(x – 2)2 + 4 b. y = (x – 4)(x + 1)

Form: Form:

Information: Information:

c. y = 2x2 + 4 d. y = -4(x + 6)2

Form: Form:


e. y = 4x2 – 3x + 8 f. y = (x – 6)2 + 1

Form: Form:


Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


35

Creating Equations Given Characteristics

You can create equations given specific characteristics such as the vertex or intercepts. Look at the problem

situation from a group of students. Analyze why they have a thumbs up or a thumbs down for the problem

below:

Write a quadratic function in factored form to represent a parabola that

opens downward and has zeros at (-4, 0) and (2, 0).

Practice: For the given characteristics, write a function to the best you can.

a. Vertex at (-5, 3) and opens down b. x-intercepts at (3, 0) and (-5, 0) and opens up

c. y-intercept at (0, 7) and opens down d. x-intercepts at (0, 0) and (2, 0) and opens down

e. Vertex at (-3, 0) and opens down f. y – intercept at (0, -2) and opens up

Sally

My function is f(x) = -(x – 2)(x + 4).

Julie

My function is f(x) = -0.5(x – 2)(x + 4).

Derek

My function is f(x) = -(x + 2)(x - 4).

Alex

My function is f(x) = 2(x – 2)(x + 4).

Monica

My function is f(x) = -2(x – 2)(x + 4).

Jeremy

My function is f(x) = ½(x – 2)(x + 4).


36

Converting between Forms

Standard to Factored – Factor your expression(factor by GCF and/or into two binomials)

a. y = x2 + 4x – 12 b. y = 3x2 – 6x

Factored to Standard – Multiply your expressions together and place in standard form. Multiply a value through

last.

a. y = (x – 3)(x + 4) b. y = 2(x – 1)(x + 2)

Vertex to Standard – Expand your squared binomial, multiply the binomials, and add constants. Multiply a

value through last.

a. y = (x – 5)2 – 12 b. y = -3(x + 1)2 + 4

Standard to Vertex - Determine your vertex (h, k) and keep the same a-value.

a. y = x2 + 4x + 3 b. y = x2 + 6x - 5


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Day 11: Average Rate of Change

Review: Find the slope (average rate of change) for the following problems:

a. b. c. (-9, 5) & (-3, 1)

When you calculate the slope of linear function, its slope is ALWAYS ______________________________.

Investigating the “Slope” of a Quadratic Function

The graph of y = -x2 + 2x + 8 is given. Fill in the table of values on the right. Then determine the slope from one

point to the next point.

What do you notice about the rate of change as you go from one point to the next?

What do you notice if you find the difference of all the slopes?

X Y

-3

-2

-1

0

1

2

3

4

5

Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


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First versus Second Differences

Quadratic Functions have constant second differences. Second differences can be calculated by finding the

rate of change with the first differences. Linear functions have constant first differences. Since quadratic

functions do not have constant first differences, they do not have a slope that remains constant for the entire

graph of a parabola.

Therefore, you are never asked to find the slope of a quadratic function, but rather the average rate of change

on a given interval. The average rate of change of a quadratic function will be different for each interval you

are asked to find, just like in your investigation problem.

Practice: For the problems below, find the average rate of change for the given intervals: Calculate average rate of change on Calculate average rate of change on

interval 0 ≤ x ≤ 2. interval 0 ≤ x ≤ 3.


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Average Rate of Change without a Graph

If you are asked to calculate the average rate of change on an interval without a graph, you will have to

come up with two points to calculate the slope.

You will get your two points by taking the bounds of your interval and substitute those x-values into your

equation to find the y-values. Then use the slope formula to calculate the slope.

Remember slope is: rise

run or 2 1

2 1

y y

x x

Practice: Calculate the average rate of change of the function y = (x – 4)2 on the given intervals:

Practice: Calculate the average rate of change of the function y = x2 + 4x – 12 on the given intervals:

2 2 x1 3 x

2 4x 3 6x


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Day 12 – Applications of the Vertex

Words that Indicate Finding Vertex

Minimum/Maximum

Minimize/Maximize

Least/Greatest

Smallest/Largest

Quadratic Equations

Standard Form: y = ax2 + bx + c y-int: (0, c)

Vertex Form: y = a(x – h)2 + k vertex: (h, k)

Factored Form: y = a(x – p)(x – q) x-int: (p, 0) & (q, 0)

Vertex: b b

,fa a2 2

Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


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1. The arch of a bridge forms a parabola modeled by the function y = -0.2(x – 40)2 + 25, where x is the horizontal

distance (in feet) from the arch’s left end and y is the corresponding vertical distance (in feet) from the base of

the arch. How tall is the arch?

2. Suppose the flight of a launched bottle rocket can be modeled by the equation y = -x2 + 6x, where y

measures the rocket’s height above the ground in meters and x represents the rocket’s horizontal distance in

meters from the launching spot at x = 0.

a. How far has the bottle rocket traveled horizontally when it reaches it maximum height? What is the

maximum height the bottle rocket reaches?

b. How far does the bottle rocket travel in the horizontal direction from launch to landing?


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3. A frog is about to hop from the bank of a creek. The path of the jump can be modeled by the equation

h(x) = -x2 + 4x + 1, where h(x) is the frog’s height above the water and x is the number of seconds since the frog

jumped. A fly is cruising at a height of 5 feet above the water. Is it possible for the frog to catch the fly, given

the equation of the frog’s jump?

4. A baker has modeled the monthly operating costs for making wedding cakes by the function

y = 0.5x2 – 12x + 150, where y is the total costs in dollars and x is the number of cakes prepared.

a. How many cakes should be prepared each month to yield the minimum operating cost?

b. What is the minimum monthly operating cost?


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5. A street vendor sells about 20 shirts a day when she charges $8 per shirt. If she decreases the price by $1, she

sells about 10 more shirts each day.

a. How many shirts does she have to sell to maximize her revenue? What is her maximum revenue?

Price Number of Shirts

Sold Revenue

$8 20

b. How much more will she make a day?

c. Write a quadratic function that models the scenario.

6. You run a canoe rental business on a small river in Georgia. You currently charge $12 per hour canoe and

average 36 rentals a day. An industry journal says that for every fifty cent increase in rental price, the average

business can expect to lose two rentals a day.

a. Use this information to attempt to maximize your income. What should you charge?

Price Number of

Rentals Revenue

$12 36

b. Write a quadratic function that models the scenario.


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Day 13 – Comparing Quadratic Functions

When comparing quadratic functions, you will want to look at their different characteristics (such as the

vertices, y-intercepts, zeros, etc). Most of the time, when you are asked to compare different quadratic

functions, they will be in different representations (table, graphs, equations, or word problems).

Example 1: Which quadratic function has the bigger y-intercept?

a. y = x2 + 4x + 7 b.

X -4 -3 -2 -1 0 1

y 0 -1 0 3 8 15

Example 2: Which quadratic functions have an x-intercept at (3, 0)?

a. y = (x + 3)(x + 1)

b.

X -1 0 1 2 3 4

y 8 3 0 -1 0 3

c.

Standard(s): _________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________

_____________________________________________________________________________________________


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Example 3: This graph shows a function f(x). Compare the graph of f(x) to the graph of the function given by

the equation g(x) = 4x2 + 6x – 18. Which function has the lesser minimum value? How do you know?

Example 4: For the function g(x) = (x – 3)2 – 2, is the average rate of change greater between x = 0 and x = 1 OR

between x = 1 and x = 2?

Unit 3B: Quadratic Functions · Algebra 1 Unit 3B: Quadratic Functions Notes 2 Day 1: Quadratic...

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