LeArning gOALS Key TermS - HHS Algebra IIheritagealgebra2.weebly.com/uploads/1/7/0/2/17020642/teachers... · groups based on these forms. The concavity of a parabola is reviewed,

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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 specific 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 r2), 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 efficient 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 r2) form.

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169D Chapter 3 Quadratic Functions

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3.1 Forms of Quadratic Functions 169

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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 specific 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

3

•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 1Students 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.

groupingHave 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?

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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

21025210215 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

2824 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.

•Have students complete Questions 2 and 3 with a partner. Then have students share their responses as a class.

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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 r2), 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.

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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 define each vocabulary word.

guiding Questions for Share Phase, Questions 2 and 3•Which form(s) of the

quadratic function is used to easily identify the y-intercept?

•Which form(s) of the quadratic function is used to easily identify if the parabola opens up or down?

•Which form(s) of the quadratic function is used to easily identify the x-intercept(s)?

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GroupingAsk a student to read the information and student work. Complete Question 4 as a class.

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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) 1 105 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 17I 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 ) 1 17 1

5 2( x 2 1 6x 1 9) 1 17 2 185 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 r2 ______

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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guiding Questions for Share Phase, Question 5•Which key characteristics

are observable when the quadratic function is written in standard form?

•Which key characteristics are observable when the quadratic function is written in factored form?

•Which key characteristics are observable when the quadratic function is written in vertex form?

•Which formulas are associated with determining the key characteristics of a quadratic function written in standard form?

•Which formulas are associated with determining the key characteristics of a quadratic function written in factored form?

•Which formulas are associated with determining the key characteristics of a quadratic function written in vertex form?

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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

21025210215 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 r2), 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 r1 1 r2 ______

2 (r1, 0), (r2, 0) Concave up when a . 0

Concave down when a , 0

Vertex y-intercept

Use r1 1 r2 ______

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

2824 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 Concave down when a , 0

Vertex y-intercept

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

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Problem 2Given 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.

groupingHave students complete Questions 1 and 2 with a partner. Then have students 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?

•Are the zeros of the quadratic 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?

•How many of the functions have 2 negative x-intercepts?

•How many of the functions have a negative y-intercept?

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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) f2(x) 5 2 1 __ 3 x2 2 3x 2 6 f3(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.

The function f2 is a possibility because it has 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 f5(x) 5 2(x 2 1)(x 2 4) f6(x) 5 2(x 2 6)2 1 3



The function f6 can be eliminated because its vertex is in Quadrant I.

f7(x) 5 23(x 1 2)(x 2 3) f8(x) 5 2(x 1 6)2 1 3

The function f7 can be eliminated because it has one positive and one negative x-intercept.

The function f8 is a possibility because it has a negative a-value and a vertex in Quadrant II.

Think about the information given by each

function and the relative position of

the graph.

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guiding Questions for Share Phase, Question 1 part (b)•Are the zeros of the quadratic




•How many of the functions have a vertex in the fourth quadrant?

•How many of the functions have 2 positive x-intercepts?

•How many of the functions have a positive y-intercept?

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b. y

x

f1(x) 5 2(x 2 75)2 2 92 f2(x) 5 (x 2 8)(x 1 2) f3(x) 5 8x2 2 88x 1 240

The function f1 is a possibility because it has a positive a-value making it concave up, and a vertex in Quadrant IV.

The function f2 can be eliminated because it does not have 2 positive x-intercepts.

The function f3 is a possibility because it has a positive a-value making it concave up, and a positive y-intercept.

f4(x) 5 23(x 2 1)(x 2 5) f5(x) 5 22(x 2 75)2 2 92 f6(x) 5 x2 1 6x 2 2

The function f4 can be eliminated because it has a negative a-value which means the graph would be concave down.

The function f5 can be eliminated because it has a negative a-value which means the graph would be concave down.

The function f6 can be eliminated because it has a negative y-intercept.

f7(x) 5 2(x 1 4)2 2 2 f8(x) 5 (x 1 1)(x 1 3)

The function f7 can be eliminated because it has a vertex in Quadrant III.

The function f8 can be eliminated because it has 2 negative x-intercepts.

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guiding Questions for Share Phase, Question 1 part (c)•Are the zeros of the quadratic




•How many of the functions have a vertex in the first quadrant?

•How many of the functions have no x-intercepts?

•How many of the functions have a positive y-intercept?

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c. y

x

f1(x) 5 3(x 1 1)(x 2 5) f2(x) 5 2(x 1 6)2 2 5 f3(x) 5 4x2 2 400x 1 10,010

The function f1 can be eliminated because it has real x-intercepts.

The function f2 can be eliminated because it has a vertex in Quadrant III.

The function f3 is a possibility because it has a positive y-intercept and a positive a-value.

f4(x) 5 3(x 1 1)(x 1 5) f5(x) 5 2(x 2 6)2 1 5 f6(x) 5 x2 1 2x 2 5

The function f4 can be eliminated because it has real x-intercepts.

The function f5 is a possibility because it has a positive a-value and a vertex in Quadrant I.

The function f6 can be eliminated because it has a negative y-intercept.

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guiding Questions for Share Phase, Question 2•What algebraic properties are

used to change the quadratic function in part (a) from factored form to standard form?

•What algebraic properties are used to change the quadratic function in part (a) from factored form to vertex form?

•What algebraic properties are used to change the quadratic 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?

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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) f1(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 r2), r1 and r2 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 axis of symmetry.

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


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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 axis of symmetry.

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


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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 3Students 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 efficient form (standard, factored, vertex) to write the function, based on the given information.

grouping•Have students complete

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?

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?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 ky 5 a(x 2 4)2 1 8y 5 2

1 _ 2 (x 2 4)2 1 8

Pat

y 5 a(x 2 h)2 1 ky 5 a(x 2 4)2 1 8y 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 r2) 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, r2, 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 determine a unique quadratic function?

•Will knowing the roots and the a-value on the function determine a unique quadratic function?

groupingAsk a student to read the information and the worked examples. Complete Questions 3 and 4 as a class.

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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? 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.

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 r2)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 field the location of the roots of the parabola?


guiding Questions for Share Phase, Question 5• 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?

•Under what circumstances is it easier to write the function in factored form?

•Under what circumstances is it easier to write the function in standard form?

• Is the maximum height of Max’s baseball the location of the vertex of the parabola?

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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? Explain your reasoning .

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 efficient 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 field . 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 4Students will write a quadratic function to represent each of three given situations.

groupingHave students complete Questions 1 through 3 with a partner. Then have students 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?

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Problem 4 Just Another Day at the Circus

Write a quadratic function to represent each situation using the given information . Be sure to define 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 fire jumper . He is attempting to run and jump through a ring of fire . He runs for 10 feet . Then, he begins his jump just 4 feet from the fire and lands on the other side 3 feet from the fire ring . When Cornelius was 1 foot from the fire 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 r2)

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 filled 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 h0

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.


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Check for Students’ Understanding

1. A catapult is designed to hurl a pumpkin from a height of 40 feet at an initial velocity of 215 feet per second.

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

h(t) 5 216t2 1 215t 1 40

b. Write an equation that you can use to determine when the pumpkin will hit the ground. Then solve the equation.

h(t) 5 216t2 1 215t 1 40

0 5 216t2 1 215t 1 40 a 5 216 b 5 215 c 5 40

t 5 2215 6 √

_________________ 2152 2 4(216)(40) __________________________

2(216)

5 2215 6 √

______________ 46,225 1 2560 _______________________

232

5 2215 6 √

_______ 48,785 ________________

232

t 5 2215 1 √

_______ 48,785 ________________

232 t 5

2215 2 √_______

48,785 ________________

232

< 20.18 < 13.62

The pumpkin will hit the ground after about 13.62 seconds.

Documents

LeArning gOALS Key TermS - HHS Algebra IIheritagealgebra2.weebly.com/uploads/1/7/0/2/17020642/teachers... · groups based on these forms. The concavity of a parabola is reviewed,