Teacher's Implementation Guide Chapter...

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169A

3.1Shape and StructureForms of Quadratic Functions

LEARNING GOALS KEY TERMS

standard form of a quadratic function

factored form of a quadratic function

vertex form of a quadratic function

concavity of a parabola

In this lesson, you will:

Match a quadratic function with its

corresponding graph.

Identify key characteristics of quadratic

functions based on the form of the function.

Analyze the different forms of

quadratic functions.

Use key characteristics of speci!c forms of

quadratic functions to write equations.

Write quadratic functions to represent

problem situations.

ESSENTIAL IDEAS

The standard form of a quadratic function is

written as f(x) 5 ax2 1 bx 1 c, where a does

not equal 0.

The factored form of a quadratic function is

written as f(x) 5 a(x 2 r1) (x 2 r

2), where a

does not equal 0.

The vertex form of a quadratic function is

written as f(x) 5 a(x 2 h)2 1 k, where a does

not equal 0.

The concavity of a parabola describes

whether a parabola opens up or opens

down. A parabola is concave down if it

opens downward, and is concave up if it

opens upward.

TEXAS ESSENTIAL KNOWLEDGE

AND SKILLS FOR MATHEMATICS

(4) Quadratic and square root functions,

equations, and inequalities. The student applies

mathematical processes to understand that

quadratic and square root functions, equations,

and quadratic inequalities can be used to

model situations, solve problems, and make

predictions. The student is expected to:

(B) write the equation of a parabola using

given attributes, including vertex, focus,

directrix, axis of symmetry, and direction

of opening

(D) transform a quadratic function

f(x) 5 a x 2 1 bx 1 c to the form

f(x) 5 a(x 2 h ) 2 1 k to identify the

different attributes of f(x)

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169B Chapter 3 Quadratic Functions

3

Overview

Students will match quadratic equations with their graphs using key characteristics. The standard form,

the factored form, and the vertex form of a quadratic equation are reviewed as is the concavity of a

parabola. Students then paste each of the functions with their graphs into one of three tables depending

on the form in which the equation is written while identifying key characteristics of each function such as

the axis of symmetry, the x-intercept(s), concavity, the vertex, and the y-intercepts. Next, a graph is

presented on a numberless coordinate plane and students identify which function(s) could model it

based on its key characteristics. Finally, a worked example shows that a unique quadratic function is

determined when the vertex and a point on the parabola are known, or the roots and a point on the

parabola are known. Students are given information about a function, and they will determine the most

ef!cient form (standard, factored, vertex) to write the function, based on the given information.

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3.1 Forms of Quadratic Functions 169C

3

Warm Up

1. Consider the quadratic functions shown.

A. f(x) 5 5x2 1 4x 1 10

B. g(x) 5 5(x 2 4) (x 1 3)

C. h(x) 5 5(x 2 3)2 1 4

a. Which function is written in vertex form? How do you know?

h(x) 5 5(x 2 3)2 1 4 is written in vertex form, because it is written in f(x) 5 a(x 2 h)2 1 k form.

b. Which function is written in standard form? How do you know?

f(x) 5 5x2 1 4x 1 10 is written in standard form, because it is written in f(x) 5 ax2 1 bx 1 c form.

c. Which function is written in factored form? How do you know?

g(x) 5 5(x 2 4)(x 1 3) is written in factored form, because it is written in f(x) 5 a(x 2 r1)(x 1 r

2) form.

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169D Chapter 3 Quadratic Functions

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3.1 Forms of Quadratic Functions 169

169

LEARNING GOALS

3.1

KEY TERMS

standard form of a quadratic function

factored form of a quadratic function

vertex form of a quadratic function

concavity of a parabola

In this lesson, you will:

Match a quadratic function with its

corresponding graph.

Identify key characteristics of quadratic

functions based on the form of the function.

Analyze the different forms of

quadratic functions.

Use key characteristics of speci!c forms of

quadratic functions to write equations.

Write quadratic functions to represent

problem situations.

Shape and StructureForms of Quadratic Functions

Have you ever seen a tightrope walker? If you’ve ever seen this, you know that it

is quite amazing to witness a person able to walk on a thin piece of rope.

However, since safety is always a concern, there is usually a net just in case of a fall.

That brings us to a young French daredevil named Phillippe Petit. Back in 1974 with

the help of some friends, he spent all night secretly placing a

450 pound cable between the World Trade Center Towers in

New York City. At dawn, to the shock and amazement of

onlookers, the fatigued 24-year old Petit stepped out onto the

wire. Ignoring the frantic calls of the police, he walked, jumped,

laughed, and even performed a dance routine on the wire for

nearly an hour without a safety net! Mr. Petit was of course

arrested upon climbing back to the safety of the ledge.

When asked why he performed such an unwise, dangerous

act, Phillippe said: “When I see three oranges, I juggle;

when I see two towers, I walk.”

Have you ever challenged yourself to do something difficult

just to see if you could do it?

You can see the events

unfold in the 2002 Academy Award winning documentary Man on Wire

by James Marsh.

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170 Chapter 3 Quadratic Functions

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What feature of the quadratic function helps determine if the parabola passes

through the origin?

What feature of the quadratic function helps determine the y-intercept?

What feature of the quadratic function helps determine the x-intercept(s)?

What feature of the quadratic function helps determine the location of

the vertex?

Problem 1

Students will match nine

quadratic functions with their

appropriate graphs using the

vertex, x-intercepts, y-intercept,

and a-value, depending on the

form of the quadratic function.

The standard form, factored

form, and vertex form of a

quadratic equation are reviewed

and students then sort their

functions and graphs into

groups based on these forms.

The concavity of a parabola is

reviewed, and students identify

the key characteristics that can

be determined from a quadratic

equation written in each form.

Students then paste each of the

functions and their graphs into

one of three tables, depending

on the form in which the

equation is written. They also

identify the axis of symmetry,

the x-intercept(s), concavity, the

vertex, and the y-intercepts for

each function.

Grouping

Have students complete

Question 1 with a partner.

Then have students share their

responses as a class.

Guiding Questions for Share Phase, Question 1

What feature of the

quadratic function helps

determine if the parabola

opens up or down?

What feature of the

quadratic function helps

determine if it has a

maximum or a minimum?

PROBLEM 1 It’s All in the Form

1. Cut out each quadratic function and graph on the next page two pages.

a. Tape each quadratic function to its corresponding graph.

Graph A, Function b Graph F, Function e

Graph B, Function a Graph G, Function i

Graph C, Function h Graph H, Function g

Graph D, Function f Graph I , Function d

Graph E, Function c

b. Explain the method(s) you used to match the functions with their graphs.

Answers will vary.

Students may identify the graphs by their vertex, x-intercept(s), y-intercept, and

a-value depending on the form of the quadratic function. They may also

substitute values of points into the functions or make a table.

Please do not use graphing

calculators for this activity. What information can you tell from looking at the function and what can you tell by looking

at each graph?

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a. f(x) 5 2(x 1 1)(x 1 5) d. f(x) 5 (x 2 1)2 g. f(x) 5 2(x 1 4)2 2 2

b. f(x) 5 1 __ 3 x2 1 πx 1 6.4 e. f(x) 5 2(x 2 1)(x 2 5) h. f(x) 5 25x2 2x 1 21

c. f(x) 5 22.5(x 2 3)(x 2 3) f. f(x) 5 x2 1 12x 2 1 i. f(x) 5 2(x 1 2)2 2 4

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A.

x0

4

22242628210

y

6

8

2

tape function here

b. f(x) 5 1 __ 3 x2 1 πx 1 6.4

B.

x0

8

2122232425

y

4

24

28

tape function here

a. f(x) 5 2(x 1 1)(x 1 5)

C.

x0

20

212122

y

30

10

210

tape function here

h. f(x) 5 25x2 2 x 1 21

D.

x0

210

25210215 5

y

220

230

240

tape function here

f. f(x) 5 x2 1 12x 2 1

E.

x0

28

216

2 422 6 8

y

224

232

tape function here

c. f(x) 5 22.5(x 2 3)(x 2 3)

F.

x0

4

422 2 6

y

8

24

28

tape function here

e. f(x) 5 2(x 2 1)(x 2 5)

G.

x0

28

24 428 8

y

216

224

232

tape function here

i. f(x) 5 2(x 1 2)2 2 4

H.

x022242628

y

22

24

26

28

tape function here

g. f(x) 5 2(x 1 4)2 2 2

I.

x0

4

2224 2 4 6

y

8

12

16

tape function here

d. f(x) 5 (x 2 1)2

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Which form(s) of the quadratic function is used to easily identify the location of

the vertex?

Which form(s) of the quadratic function is used to easily identify the axis

of symmetry?

Grouping

Ask a student to read the

information. Discuss

as a class.


Questions 2 and 3 with a

partner. Then have students

share their responses

as a class.

Recall that quadratic functions can be written in different forms.

standard form: f(x) 5 ax2 1 bx 1 c, where a does not equal 0.

factored form: f(x) 5 a(x 2 r1)(x 2 r

2), where a does not equal 0.

vertex form: f(x) 5 a(x 2 h)2 1 k, where a does not equal 0.

2. Sort your graphs with matching equations into 3 piles

based on the function form.

The graphs of quadratic functions can be described using

key characteristics:

x-intercept(s),

y-intercept,

vertex,

axis of symmetry, and

concave up or down.

Concavity of a parabola describes whether a parabola opens up or opens down.

A parabola is concave down if it opens downward; a parabola is concave up if it

opens upward.

3. The form of a quadratic function highlights different key characteristics.

State the characteristics you can determine from each.

standard form

I can determine the y-intercept, and whether the parabola is concave up or down

when the quadratic is in standard form.

factored form

I can determine the x-intercepts, and whether the parabola is concave up or

concave down when the quadratic is in factored form.

vertex form

I can determine the vertex, whether the parabola is concave up or concave down,

and the axis of symmetry when the quadratic is in vertex form.

Keep these piles; you will use

them again at the end of this Problem.

Make a list of words used

to describe quadratic

functions: x-intercept,

y-intercept, vertex, axis of

symmetry, and concave

up or down. Pronounce

each word aloud, having

students repeat after

you. Draw a quadratic

function graph on the

board. Use the graph

to help de!ne each

vocabulary word.

Make a lis

Guiding Questions for Share Phase, Questions 2 and 3

Which form(s) of the

quadratic function is

used to easily identify the

y-intercept?


quadratic function is used to

easily identify if the parabola

opens up or down?


quadratic function is

used to easily identify the

x-intercept(s)?

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Grouping


information and student work.

Complete Question 4 as

a class.

4. Christine, Kate, and Hannah were asked to determine the vertex of three different

quadratic functions each written in different forms. Analyze their calculations.

Christine

f(x) 5 2x2 1 12x 1 10

The quadratic function is in standard

form. So I know the axis of symmetry

is x 5 2b ___ 2a .

x 5 212 ____

2(2)

5 23.

Now that I know the axis of symmetry,

I can substitute that value into the

function to determine the

y-coordinate of the vertex.

f(23) 5 2(23)2 1 12(23) 1 10

5 2(9) 2 36 1 10

5 18 2 36 1 10

5 8

Therefore, the vertex is (23, 8).

Kate

g(x) 5 1 _ 2 (x 1 3)(x 2 7)

The form of the function tells me the x-intercepts are 23 and 7. I also know the x-coordinate of the vertex will be directly in the middle of the x-intercepts. So, all I have to do is calculate the average.

x 5 23 1 7 _______ 2

5 4 __ 2 5 2

Now that I know the x-coordinate of the vertex, I can substitute that value into the function to determine the y-coordinate.

g(2) 5 1 _ 2 (2 1 3)(2 2 7)

5 1 _ 2 (5)(25)

5 212.5

Therefore, the vertex is (2, 212.5).

Hannah

h(x) 5 2 x 2 1 12x 1 17

I can determine the vertex by rewriting the function in vertex form.To do that, I need to complete the square.

h(x) 5 2 x 2 1 12x 1 17

5 2( x 2 1 6x 1 11 ) 1 17 1 1 1

5 2( x 2 1 6x 1 9) 1 17 2 18

5 2(x 1 3 ) 2 2 1

Now, I know the vertex is (23, 21).

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a. How are these methods similar? How are they different?

Christine’s method and Kate’s method require that you determine the axis of

symmetry, and then substitute that value into the function to determine the

y-coordinate of the vertex.

Their methods are different in the way the axis of symmetry was determined.

Christine used x 5 2b ____

2a and Kate used x 5

r1 1 r

2 ______ 2 .

Hannah completed the square to rewrite her equation in vertex form. When a

quadratic equation is in vertex form, f(x) 5 a(x 2 h ) 2 1 k, the coordinates of the

vertex are (h, k).

b. What must Kate do to use Christine’s method?

Kate knows the a-value from the form of her quadratic equation. She must

multiply the factors together and combine like terms. She would then have a

quadratic function in standard form to determine the b-value.

c. What must Christine do to use Kate’s method?

Christine must factor the quadratic function or use the quadratic formula to

determine the x-intercepts. Once she determines the x-intercepts, she can use

the same method as Kate.

d. Describe the steps Hannah used to complete the square and rewrite her equation in

vertex form.

To complete the square and rewrite her equation in vertex form, Hannah

completed the following steps:

Factor out a 2 from 2 x 2 1 12x.

Complete the square by adding 9 to x 2 1 6x. She calculated 9 by dividing the

coefficient of 6x by 2, then squaring the result, ( 6 __ 2 )

2

.

Add 218 to maintain balance in the equation. Adding 9 to x 2 1 6x results in

adding 18 to the equation because the quantity ( x 2 1 6x 1 9) is multiplied by 2.

Adding 218 maintains balance in the equation.

Rewrite x 2 1 6x 1 9 in factored form, (x 1 3 ) 2 , and subtract 17 218.

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Grouping






Which key characteristics

are observable when the

quadratic function is written

in standard form?




in factored form?




in vertex form?

Which formulas are

associated with determining

the key characteristics of a

quadratic function written in

standard form?

Which formulas are




factored form?

Which formulas are




vertex form?

5. Analyze each table on the following three pages. Paste each function and its

corresponding graph from Question 2 in the “Graphs and Their Functions” section of

the appropriate table. Then, explain how you can determine each key characteristic

based on the form of the given function.

Standard Form

f(x) 5 ax2 1 bx 1 c, where a fi 0

Graphs and Their Functions

A.

x0

4

22242628210

y

6

8

2

b. f(x) 5 1 __ 3 x2 1 πx 1 6.4

C.

x0

20

212122

y

30

10

210

h. f(x) 5 25x2 2 x 1 21

D.

x0

210

25210215 5

y

220

230

240

f. f(x) 5 x2 1 12x 2 1

Methods to Identify and Determine Key Characteristics

Axis of Symmetry x-intercept(s) Concavity

x 5 2b ____

2a Substitute 0 for y, and then solve

for x using the quadratic formula,

factoring, or a graphing

calculator.

Concave up when a . 0

Concave down when a , 0

Vertex y-intercept

Use 2b ____

2a to determine the x-coordinate of the

vertex. Then substitute that value into the

equation and solve for y.

c-value

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Factored Form

f(x) 5 a(x 2 r1)(x 2 r

2), where a fi 0


B.

x0

8

2122232425

y

4

24

28

a. f(x) 5 2(x 1 1)(x 1 5)

E.

x0

28

216

2 422 6 8

y

224

232

c. f(x) 5 22.5(x 2 3)(x 2 3)

F.

x0

4

422 2 6

y

8

24

28

e. f(x) 5 2(x 2 1)(x 2 5)



x 5 r

1 1 r

2 ______ 2

(r1, 0), (r

2, 0) Concave up when a . 0


Vertex y-intercept

Use r

1 1 r

2 ______ 2 to determine the x-coordinate of the

vertex. Then substitute that value into the equation

and solve for y.

Substitute 0 for x, and then solve for y.

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Vertex Form

f(x) 5 a(x 2 h)2 1 k, where a fi 0


G.

x0

28

24 428 8

y

216

224

232

i. f(x) 5 2(x 1 2)2 2 4

H.

x022242628

y

22

24

26

28

g. f(x) 5 2(x 1 4)2 2 2

I.

x0

4

2224 2 4 6

y

8

12

16

d. f(x) 5 (x 2 1)2



x 5 h Substitute 0 for y, and then solve

for x using the quadratic formula,

factoring, or a graphing

calculator.

Concave up when a . 0


Vertex y-intercept

(h, k) Substitute 0 for x, and then solve for y.

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Problem 2

Given a graph on a numberless

coordinate plane, students will

identify functions that model the

graph. Next, they identify the

form of the function, and rewrite

the function in the other two

forms when possible.

Grouping


Questions 1 and 2 with a


share their responses as

a class.

Guiding Questions for Share Phase, Question 1 part (a)

Are the zeros of the quadratic

function real or imaginary?

How do you know?


function negative or positive

or both? How do you know?

Is the a-value of the

quadratic function

negative or positive?

How do you know?

How many of the functions

have a vertex in the

second quadrant?


have 2 negative x-intercepts?


have a negative y-intercept?

PROBLEM 2 What Do You Know?

1. Analyze each graph. Then, circle the function(s) which could model the graph.

Describe the reasoning you used to either eliminate or choose each function.

a.

x

y

f1(x) 5 22(x 1 1)(x 1 4) f

2(x) 5 2

1 __

3 x2 2 3x 2 6 f

3(x) 5 2(x 1 1)(x 1 4)

The function f1 is a

possibility because it has

a negative a-value and 2

negative x-intercepts.



a negative a-value and a

negative y-intercept.

The function f3 can be

eliminated because it has

a positive a-value which

means the graph would

be concave up.

f4(x) 5 2x2 2 8.9 f

5(x) 5 2(x 2 1)(x 2 4) f

6(x) 5 2(x 2 6)2 1 3





be concave up.





be concave up.


eliminated because its

vertex is in Quadrant I.

f7(x) 5 23(x 1 2)(x 2 3) f

8(x) 5 2(x 1 6)2 1 3



one positive and one

negative x-intercept.



a negative a-value and a

vertex in Quadrant II.

f7ff

The

eli

one

Think about the information given by each

function and the relative position of

the graph.one pos

negati

one

neg

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Guiding Questions for Share Phase, Question 1 part (b)



How do you know?





quadratic function


How do you know?

How many of the

functions have a vertex

in the fourth quadrant?


have 2 positive x-intercepts?


have a positive y-intercept?

b. y

x

f1(x) 5 2(x 2 75)2 2 92 f

2(x) 5 (x 2 8)(x 1 2) f

3(x) 5 8x2 2 88x 1 240



a positive a-value making

it concave up, and a

vertex in Quadrant IV.


eliminated because it

does not have 2 positive

x-intercepts.



a positive a-value making

it concave up, and a

positive y-intercept.

f4(x) 5 23(x 2 1)(x 2 5) f

5(x) 5 22(x 2 75)2 2 92 f

6(x) 5 x2 1 6x 2 2



a negative a-value which


be concave down.



a negative a-value which


be concave down.



a negative y-intercept.

f7(x) 5 2(x 1 4)2 2 2 f

8(x) 5 (x 1 1)(x 1 3)



a vertex in Quadrant III.



2 negative x-intercepts.

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Guiding Questions for Share Phase, Question 1 part (c)



How do you know?





quadratic function


How do you know?

How many of the

functions have a vertex

in the #rst quadrant?


have no x-intercepts?


have a positive y-intercept?

c. y

x

f1(x) 5 3(x 1 1)(x 2 5) f

2(x) 5 2(x 1 6)2 2 5 f

3(x) 5 4x2 2 400x 1 10,010



has real x-intercepts.



a vertex in Quadrant III.



a positive y-intercept and a

positive a-value.

f4(x) 5 3(x 1 1)(x 1 5) f

5(x) 5 2(x 2 6)2 1 5 f

6(x) 5 x2 1 2x 2 5



has real x-intercepts.



a positive a-value and a

vertex in Quadrant I.



a negative y-intercept.

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What algebraic properties are

used to change the quadratic

function in part (a) from

factored form to

standard form?



function in part (a) from

factored form to vertex form?



function in part (b) from

standard form to vertex form?

Why can’t the quadratic

function in part (b) be written

in factored form?

What algebraic properties

are used to change the

quadratic function in

part (c) from vertex form to

standard form?

Why can’t the quadratic

function in part (c) be written

in factored form?

2. Consider the three functions shown from Question 1.

Identify the form of the function given.

Write the function in the other two forms, if possible. If it is not possible, explain why.

Determine the y-intercept, x-intercepts, axis of symmetry, vertex, and concavity.

a. From part (a): f1(x) 5 22(x 1 1)(x 1 4)

The function is given in factored form.

Standard Form: Vertex Form:

f1(x) 5 22(x 1 1)(x 1 4) f

1(x) 5 22(x2 1 5x 1 4)

5 22(x2 1 5x 1 4) 5 22 ( x2 1 5x 1 25 ___ 4 ) 1 9 __

2

5 22x2 2 10x 2 8 5 22 ( x 1 5 __ 2 )

2

1 9 __ 2

The y-intercept is (0, 28). In standard form, f(x) 5 ax2 1 bx 1 c, c represents

the y-intercept.

The x- intercepts are (2 4, 0) and (2 1, 0). In factored form, f(x) 5 a(x 2 r1)(x 2 r

2),

r1 and r

2 represent the x-intercepts.

The axis of symmetry is x 5 2 5

__ 2 . In vertex form, f(x) 5 a(x 2 h)2 1 k, h represents

the axis of symmetry.

The vertex ( 2 5

__ 2 , 9 __

2 ) . In vertex form, f(x) 5 a(x 2 h)2 1 k, (h, k) represents the vertex.

The parabola is concave up because the value of a is positive in vertex form

f(x) 5 a(x 2 h)2 1 k.

b. From part (c): f3(x) 5 4x2 2 400x 1 10,010

The function is given in standard form.

Vertex Form: Factored Form:

f3(x) 5 4x2 2 400x 1 10,010 Answers will vary.

5 4(x2 2 100x 1 2500) 1 10,010 2 10,000 The function does not cross the

x-axis, therefore it does not have

real number x-intercepts. I cannot

factor this function.

5 4(x 2 50)2 1 10

The y- intercept is (0, 10,010). In standard form, f(x) 5 ax2 1 bx 1 c,

c represents the y- intercept.

There are no real x- intercepts because I know the function does not cross the

x-axis.

The axis of symmetry is x 5 50. In vertex form, f(x) 5 a(x 2 h)2 1 k , h represents


The vertex is (50, 10). In vertex form, f(x) 5 a(x 2 h)2 1 k, (h, k) represents the

vertex.


f(x) 5 a(x 2 h)2 1 k.

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c. From part (c): f5(x) 5 2(x 2 6)2 1 5

The function is given in vertex form.

Standard Form: Factored Form:

f5(x) 5 2_x 2 6+2 1 5 Answers will vary.

5 2_x2 2 12x 1 36+ 1 5 The function does not cross the

x-axis, therefore it does not have

real number x-intercepts. I cannot

factor this function.

5 2x2 2 24x 1 72 1 5

5 2x2 2 24x 1 77

The y- intercept is (0, 77). In standard form, f(x) 5 ax2 1 bx 1 c, c represents the

y- intercept.

There are no real x- intercepts because I know the function does not cross the

x-axis.

The axis of symmetry is x 5 6. In vertex form, f(x) 5 a(x 2 h)2 1 k , h represents


The vertex is (6, 5). In vertex form, f(x) 5 a(x 2 h)2 1 k, (h, k) represents the vertex.


f(x) 5 a(x 2 h)2 1 k.

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If a quadratic function is written in factored form, and the roots are given,

which variables are known?

How many different quadratic functions have the same vertex?

How many different quadratic functions have the same zeros?

Will knowing the vertex and the zeros determine a unique quadratic function?

Will knowing the vertex and the a-value determine a unique quadratic function?

Problem 3

Students will explore the

number of unknowns when

quadratic functions are

written in the different forms.

A worked example shows that

a unique quadratic function is

determined when the vertex

and a point on the parabola are

known, or the roots and a point

on the parabola are known.

In the last activity, students

are given information about

the function, and they then

determine the most ef#cient

form (standard, factored, vertex)

to write the function, based on

the given information.

Grouping


Question 1 on their own

and discuss.

Complete Question 2 as

a class.

Guiding Questions for Discuss Phase, Questions 1 and 2 part (c)

George wrote his quadratic

equation using which form?

Pat wrote her quadratic

equation using which form?

Do both quadratic equations

have a vertex at (4, 8)?

How many different

parabolas could have a

vertex at (4, 8)?

If a quadratic function is

written in vertex form, and

the vertex is given, which

variables are known?

?PROBLEM 3 Unique . . . One and Only

1. George and Pat were each asked to write a quadratic equation with a vertex of (4, 8).

Analyze each student’s work. Describe the similarities and differences in their

equations and determine who is correct.

George

y 5 a(x 2 h)2 1 k

y 5 a(x 2 4)2 1 8

y 5 2 1 _ 2 (x 2 4)2 1 8

Pat

y 5 a(x 2 h)2 1 k

y 5 a(x 2 4)2 1 8

y 5 (x 2 4)2 1 8

Both George and Pat are correct.

George and Pat each used the vertex form of a quadratic equation and

substituted h 5 4 and k 5 8. George chose a 5 2 1

__ 2 and Pat chose a 5 1.

There was not information given to create a unique quadratic equation, therefore,

both equations represent a quadratic equation with the vertex (4, 8).

2. Consider the 3 forms of quadratic functions and state the number of unknown values

in each.

FormNumber of Unknown

Values

f(x) 5 a(x 2 h)2 1 k 5

f(x) 5 a(x 2 r1)(x 2 r

2) 5

f(x) 5 ax2 1 bx 1 c 5

a. If a function is written in vertex form and you know the vertex, what is still unknown?

I still have 3 unknowns: x, y, and a.

b. If a function is written in factored form and you know the roots, what is

still unknown?

I still have 3 unknowns: x, y, and a.

c. If a function is written in any form and you know one point, what is still unknown?

State the unknown values for each form of a quadratic function.

If the function is written in vertex form, I still have 3 unknowns: a, h, and k.

If the function is written in factored form, I still have 3 unknowns: r1, r

2, and a.

If the function is written in standard form, I still have 3 unknowns: a, b, and c.

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Guiding Questions for Discuss Phase, Question 2 parts (d) and (e)

Will knowing the vertex

and a point on the function

determine a unique quadratic

function?

Will knowing the roots and

a point on the function


function?

Will knowing the roots and

the a-value on the function


function?

Grouping


information and the worked

examples. Complete

Questions 3 and 4 as a class.

d. If you only know the vertex, what more do you need to write a unique function?

Explain your reasoning.

I will need a point or the a-value. If I have a point I can solve for the a-value, if I

have the a-value, then I have the unique equation.

e. If you only know the roots, what more do you need to write a unique function?


I will need a point or the a-value. If I have a point I can solve for the a-value, if I

have the a-value, then I have the unique equation.

You can write a unique quadratic function given a vertex and a point on the parabola.

Write the quadratic function given the vertex (5, 2) and the point (4, 9).

f(x) 5 a(x 2 h)2 1 k

9 5 a(4 2 5)2 1 2

9 5 a(21)2 1 2

9 5 1a 1 2

7 5 1a

7 5 a

f(x) 5 7(x 2 5)2 1 2

Substitute the given values into

the vertex form of the function.

Then simplify.

Finally, substitute the a-value

into the function.

You can write a unique quadratic function given the roots and a point on the parabola.

Write a quadratic function given the roots (22, 0) and (4, 0), and the point (1, 6).

f(x) 5 a(x 2 r1)(x 2 r

2)

6 5 a(1 2 (22))(1 2 4)

6 5 a(1 1 2)(1 2 4)

6 5 a(3)(23)

6 5 29a

2 2 __ 3 5 a

f(x) 5 2 2 __

3 (x 1 2)(x 2 4)

Substitute the given values into

the factored form of the function.

Then simplify.

Finally, substitute the a-value

into the function.

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Are the 35 and 38 yard lines on the football #eld the location of the roots of

the parabola?

Grouping






If you know the minimum

or maximum point of a

quadratic function, is that

always the vertex of the

function?

Under what circumstances is

it easier to write the function

in vertex form?

If you know three points on

the quadratic function, is it

always easier to write the

function in factored form?



in factored form?



in standard form?

Is the maximum height of

Max’s baseball the location

of the vertex of the parabola?

3. Explain why knowing the vertex and a point creates a unique quadratic function.

A unique quadratic function is created because 4 of the 5 unknowns are given, which

means there is only one possible a-value.

4. If you are given the roots, how many unique quadratic functions can you write?


I can write an infinite number of quadratic functions. If I am only given the

roots, I can assign any a-value that I want.

5. Use the given information to determine the most ef!cient form you could use to

write the function. Write standard form, factored form, vertex form, or none in the

space provided.

a. minimum point (6, 275) vertex form

y-intercept (0, 15)

b. points (2, 0), (8, 0), and (4, 6) factored form

c. points (100, 75), (450, 75), and (150, 95) standard form

d. points (3, 3), (4, 3), and (5, 3) none

e. x-intercepts: (7.9, 0) and (27.9, 0) factored form

point (24, 24)

f. roots: (3, 0) and (12, 0) factored form

point (10, 2)

g. Max hits a baseball off a tee that is 3 feet high. vertex form

The ball reaches a maximum height of 20 feet

when it is 15 feet from the tee.

h. A grasshopper was standing on the 35 yard factored form

line of a football !eld. He jumped, and landed

on the 38 yard line. At the 36 yard line he was

8 inches in the air.

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What distance has Crazy Cornelius traveled when he reaches a height of 3.5 feet?

Did you write the quadratic function in factored form?

If Harsh Knarsh begins on a ramp 30 feet high, is this associated to the

y-intercept of the graph?

What function or equation is associated with vertical motion?

Is the equation for vertical motion written in standard form?

Problem 4

Students will write a quadratic

function to represent each of

three given situations.

Grouping


Questions 1 through 3 with a


share their responses as a class.

Guiding Questions for Share Phase, Questions 1 through 3

Which key characteristic is

associated with the given

information?

If Amazing Larry reaches a

maximum height of 30 feet,

will this be the vertex of

the graph?

Did you write the quadratic

function in vertex form?

If the cannon is 10 feet above

the ground, is this associated

with the y-intercept of

the graph?

What is the concavity of

the parabola? How do

you know?

Is the a-value of the function

a positive number or a

negative number?

What distance has Crazy

Cornelius traveled before he

leaves the ground?

What distance has Crazy

Cornelius traveled as he

lands back on the ground?

Are the points or distances

at which Crazy Cornelius

lifts off the ground and lands

back down on the ground

associated with the roots of

the function or x-intercepts?

PROBLEM 4 Just Another Day at the Circus

Write a quadratic function to represent each situation using the given information. Be sure to

de!ne your variables.

1. The Amazing Larry is a human cannonball. He would like to reach a maximum height of

30 feet during his next launch. Based on Amazing Larry’s previous launches, his

assistant DaJuan has estimated that this will occur when he is 40 feet from the cannon.

When Amazing Larry is shot from the cannon, he is 10 feet above the ground. Write a

function to represent Amazing Larry’s height in terms of his distance.

Let h(d) represent Amazing Larry’s height in terms of his distance, d.

h(d) 5 a(d 2 40)2 1 30

10 5 a(0 2 40)2 1 30

10 5 1600a 1 30

220 5 1600a

2 1 ___ 80

5 a

h(d) 5 2 1 ___ 80

(d 2 40)2 1 30

2. Crazy Cornelius is a !re jumper. He is attempting to run and jump through a ring of !re.

He runs for 10 feet. Then, he begins his jump just 4 feet from the !re and lands on the

other side 3 feet from the !re ring. When Cornelius was 1 foot from the !re ring at the

beginning of his jump, he was 3.5 feet in the air. Write a function to represent Crazy

Cornelius’ height in terms of his distance. Round to the nearest hundredth.

Let h(d) represent Crazy Cornelius’s height in terms of his distance, d.

h(d) 5 a(d 2 r1)(d 2 r

2)

3.5 5 a(13 2 10)(13 2 17)

3.5 5 a(3)(24)

3.5 5 212a

20.29 5 a

h(d) 5 20.29(d 2 10)(d 2 17)

3. Harsh Knarsh is attempting to jump across an

alligator !lled swamp. She takes off from a

ramp 30 feet high with a speed of 95 feet

per second. Write a function to represent

Harsh Knarsh’s height in terms of time.

Let h(t) represent Harsh Knarsh’s height in terms of his time, t.

h(t) 5 216t2 1 v0t 1 h

0

h(t) 5 216t2 1 95t 1 30

Be prepared to share your solutions and methods.

Remember, the general equation to

represent height over time is h(t) 5 216t2 1 v

0t 1 h

0 where

v0 is the initial velocity in feet per second and h

0 is the

initial height in feet.

time, t.

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