Name_
Class
Date
Graphing Quadratic Functions
in Vertex Form
COMMON
t.. CORE
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CC.9-12.A.CED.2",
CC.9-12.FIF.2,
Essential question: How can you graph the function f(x) = a(x - h) 2 + lc?
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ENGAGE
CC.9-121.1F.7a*,
CC.9-I2.F.BF 3
Understanding Vertex Form
The vertex form of a quadratic function is f(x) = a(x - h) 2 + k. The vertex of the graph
of a quadratic function in vertex form is (h, k).
f(x) = a(x - it) 2 + k
a indicates a vertical
stretch or shrink and/or
a reflection across the
x-axis.
translation.
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REFLECT \
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1a. For the functionfix) = 2(x - 3) 2 + 1, what
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Th x-coordinates
ar zeros.
x
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are the values of a, h, and k? What do each
of these values indicate about the graph of
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A zero Of a function is an input value x that
makes the output value f(x) equal 0. You can estimate
the zeros of a quadratic function by observing where
the graph crosses the x-axis.
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k indicates a
vertical translation.
h indicates a horizontal
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the function?
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1b. Explain why the vertex of the graph of a quadratic function in vertex form is (h, k).
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lc. If you estimate a zero of a quadratic function from a graph, how could you use
algebra to check your answer?
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Lesson 3
EXAMPLE Graphing f(x).= a(x — h) 2 + k
Graph the function f(x) = 2(x + 1) 2 - 2. Identify the vertex, minimum or
maximum, axis of symmetry, and zeros of the function.
A Identify and graph the vertex.
h=k= - :1)
The vertex of the graph is
B Identify the coordinates of points to the left and right of the vertex.
x
—3
—2
0
1
f (x)
C:,
0
0
Co
C Graph the points and connect them with a smooth curve.
•
Identify the minimum or maximum.
The graph opens upward, so the function has a
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The minimum is
E Identify the axis of symmetry.
Identify the zeros of the function.
The graph appears to cross the x-axis at the points ( —
,
/2)_ and
so the zeros of the function appear to be — a
and
°
REFLECT
2a.
Flow could you use the value of a to determine whether the function
f(x) = 2(x -F. 1)2 — 2 has a minimum or a maximum?
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The axis of symmetry is the vertical line x =
2b. How could you use the table in part B to confirm that you correctly identified the
zeros of the function from its graph?
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Lesson 3
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EXAMP-MLE Writing Equations in Vertex Form
Write the vertex form of the quadratic function whose graph is shown.
A Use the vertex of the graph to identify the values of h and k.
The vertex of the graph is ( 0 1 0 )
h=
k=
a
Substitute the values of h and k into the vertex form:
2
a) +
f(x) = aCx —
Use the point ( 2,
-
—
6) to identify the value of a.
f(x) = a(x - 2)2 + 2
—
= a( -
—21 +
I
—
Vertex form
2
Substitute
2
—
6 for f(x) and
—
2 for x.
Simplify.
2
= a(16)
Subtract 2 from both sides.
=a
Divide both sides by 16.
Substitute the value of a into the vertex form:
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+2
So, the vertex form of the function shown in the graph is
[ REFLECT
3a. How can you tell by looking at the graph that the value of a is negative?
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3b. Describe the graph of the given function as a transformation of the parent
quadratic function.
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Lesson 3
EXAMPLE\ Modeling Quadratic Functions in Vertex Form
The shape of a bridge support can be modeled by f(x)
= — 601 0 x —
300)2 + 150,
where x is the horizontal distance in feet from the left end of the bridge and
f(x) is the height in feet above the bridge deck. Sketch a graph of the support.
Then determine the maximum height of the support above the bridge deck and
the width of the support at the level of the bridge deck.
A
Graph the function.
• The vertex of the graph is
(3 DO ) ISO)
• Find the point at the left end of the support (x •-= 0).
Sincef(0) —
0
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the point
03_ _ represents the left end.
• Use symmetry to find the point at the right end of the support.
Since the left end is 300 feet to the left of the vertex, the right end will be 300 feet to
the right of the vertex.
The point (
0° ) 6) represents the right end.
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• Find two other points on the support.
(120, Ot ) and (480,
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• Sketch the graph.
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Determine the maximum height of the support.
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So, the maximum height of the bridge support is
the width of the bridge support at the level of the bridge deck.
The distance from the left end to the right end is
So, the width is (0 0
C
feet.
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The maximum of the function is _ 150
feet.
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4a. Explain how you know that the y-coordinate of the right end of the support is 0.
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Graph each quadratic function. Identify the vertex, minimum or maximum, axis
of symmetry, and zeros of the function.
1. f(x) = —2x2 +
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Write the vertex form of each quadratic function.
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Lesson 3
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7. The functionf(x) = —16(x — 1) 2 + 16 gives the height in feet
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of a football x seconds after it is kicked from ground level.
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a. Sketch a graph of the function.
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b. What is the maximum height that the ball reaches?
IC -14c-4C. How long does the ball stay in the air? Explain how you
determined your answer.
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8. A technician is launching an aerial firework from a
tower. The height of the firework in feet is modeled by
the functionf(x)= —16(x — 3) 2 + 256 where x is the time
in seconds after the firework is launched.
a. Sketch a graph of the function.
b. Professional fireworks are usually timed to explode
as they reach their highest point. How high will the
firework be when it reaches its highest point?
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firework is launched? Explain how you determined
your answer.
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minimum: a function whose graph has
a vertex at (-5, —1) or the function
graphed below?
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10. Which quadratic function has a lesser
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or the function graphed below?
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9. Which quadratic function has a greater
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