Download Writing Quadratic Functions in Vertex F rrn

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Name
Class
Date
Writing Quadratic Functions
in Vertex F rrn
: COMMON
CORE
••.......*
CC.9-12ASSE.3b*,
CC.9-12ACED.2*,
CC.9-121.1F.2,
CC.9-12.EIE7*.
CC.9-121.IE7a*,
CC.9-12.1.1F.8a
Essential question: How do you convert quadratic functions to vertex form
f(x) = a(x — h) 2 + k?
Writing Quadratic Functions in Different Forms
Write the quadratic function in the form described.
A Write the functionf(x) = 2(x
—
4)2 + 3 in the form f(x) = ax2 + bx + c.
fix) = 2(x — 4) 2 + 3
f(x) = 2(2 - - 1 )( + Iç + 3
Multiply to expand (x — 4) 2 .
f(x)
Distribute 2.
2(x2) —
.f(x) = 212 -
(
8x) +
(16) + 3
K + 3) + 3
Multiply.
..f(x) = 2? - 16x + '35
Combine like terms.
-
f
So, fix) = 2(x — 4)2 ±
°C
3 is equivalent to /61X
C
Write the function f(x) = 12 + 6x + 4in vertex form.
Recall that the vertex form of a quadratic function is f(x) = a(x h)2 4 k.
Write the given function in vertex form by completing the square.
Ci Houg hto n Mifflin Harcourt Pu blishing Co mpany
—
f(x) = x2 + 6x +4
Set up for completing the square.
f(x) = (x2 + 6x + I
)+4—Li
Ax) = (12 + 6x + 9) + 4 — 9
Add a constant so the expression inside
the parentheses is a perfect square
trinomial. Subtract the constant to
keep the equation balanced.
2
f(x) =(x +
Write (x2 + 6x + 9) as a binomial
squared.
fix). (x + 3)2 - 5
Combine like terms.
z
Sodix) = x2 6x + 4 is equivalent to
—
5
REFLECT \
1 a. In part A, how does the value of a of the function in vertex form compare with the
value of a when the function is in the form fix) = ax2 + bx c?
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lb. In part B, how could you check that you found the vertex form of the quadratic
equation correctly?
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lc. Describe how to complete the square for the quadratic expression
x2 +8x+ 1
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The standard form of a quadratic equation isf(x) = ax2 + bx + c. Any quadratic function
in standard form can be written in vertex form, and any quadratic function in vertex form
can be written in standard form.
EXAMPLE\ Graphing by Completing the Square
Graph the function by first writing it in vertex form. Then give the maximum or
minimum of the function and identify its zeros.
A
• Write the function in vertex form.
Set up for completing the square.
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= (X - 9
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/Cr)
Add a constant to complete the
square. Subtract the constant to keep
the equation balanced.
2
Write the expression in parentheses
as a binomial squared.
fix) (x - 4)2 - tj
Combine like terms.
• Sketch a graph of the function.
The vertex is 1 Li
1.6
Two points to the left of the vertex are
(2,
D ) and (3, - 3 ).
Two points to the right of the vertex are
(5, -
) and (6, Q)•
• Describe the function's properties.
The minimum is _ _
The zeros are
Unit 2
and
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48
Lesson 4
B f(x) = -2x2 - 12x - 16
• Write the function in vertex form.
f(x) =
(x2 + 6x) -
16
Factor the variable terms so
that the coefficient of x2 is 1.
f(4=
Set up for completing the
square.
16 -
9)—
16 — (-2)
9
Complete the square. Since
the constant is multiplied
by -2, subtract the product
of -2 and the constant to
keep the equation balanced.
5
Write the expression in
parentheses as a binomial
squared.
f(x) = -2(x + 3) 2 - 16- ( -
Simplify (-2)9.
Combine like terms.
• Sketch a graph of the function.
The vertex is
Two points to the left of the vertex are
(-5, -ç ) and (-4, C
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Two points to the right of the vertex are
(-2:
) and (-1, - )•
• Describe the function's properties.
The maximum is
The zeros are —
[REFLECT
and —
N
2a. How do you keep the equation of a quadratic function balanced when completing
the square?
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the square?
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2c. Why might the vertex form of a quadratic equation be more useful in some
situations than the standard form?
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XAMPLE
Modeling Quadratic Functions in Standard Form
The function h(t) = —16t2 + 64t gives the height h in feet of a golf ball t seconds
after it is hit. The ball has a height of 48 feet after 1 second. Use the symmetry
of the function's graph to determine the other time at which the ball will have a
height of 48 feet.
A
Write the function in vertex form.
— 4t)
Factor so that the coefficient
of t2 is 1.
— -16) LI
kt) —16(1 2 — 4t +
Complete the square and
keep the equation balanced.
2
/KO= - 1+ —
Write the expression in
parentheses as a binomial
squared.
h(t) = —16(t— 2) 2 +
4
Simplify.
This point is 2 units to the left of the vertex. Based on
symmetry, there is a point 2 units to the right of the
vertex with the same y-coordinate at ( L.I
KY
:E "0
The point (0, 0 is on the graph.
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The vertex is
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D, 3
1-1- me (5)
The point (1, 48) is on the graph. Based on symmetry,
the point( 3 , 48) is also on th
e graph.
So, the ball will have a height of 48 feet after 1 second and again after
3
seconds.
RE'iLECT
3a.
How can you check your answer to the problem?
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Use symmetry to sketch a graph of the function and solve the
problem.
3b. What is the maximum height that the ball reaches? How do you know?
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3c. If you know the coordinates of a point to the left of the vertex of the graph of a
quadratic function, how can you use symmetry to find the coordinates of another
point on the graph?
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Graph each function by first writing it in vertex form. Then give the maximum
or minimum of the function and identify its zeros.
1. f(x) = x,21
—
fix + 9
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—
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k(x-3 3
3. f(x) =
2.
7x2
-
m
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4.
fix)
arc) 3
= 3x2 — 12x + 9
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5. A company is marketing a new toy. The function s(p) =
—
50/92 + 3000p models
how the total sales $ of the toy, in dollars, depend on the price p of the toy,
in dollars.
a. Complete the square to write the function in vertex
form and then graph the function.
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35, 000
b. What is the vertex of the graph of the function?
What does the vertex represent in this situation?
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The model predicts that total sales will be $40,000 when the toy price is $20.
At what other price does the model predict that total sales will be $40,000?
Use the symmetry of the graph to support your answer.
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6. A circus performer throws a ball from a height of 32 feet. The model
h(0 = —16? + 16t + 32 gives the height of the ball in feet 1 seconds
after it is thrown.
a. Complete the square to write the function in vertex form
and then graph the function.
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b. What is the maximum height that the ball reaches?
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What is a reasonable domain of the function? Explain.
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d. What is the y-intercept of the function's graph? What does it represent in this
situation? What do you notice about they-intercept and the value of
c when the function is written in standard form?
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