Quadratic Functions

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Quadratic Functions

• Vertex Form

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Example 1
Use the description to write the quadratic
function in vertex form.
The parent function f(x) x2 is vertically
stretched by a factor of and then
translated 2 units left and 5 units down to
create g.
Step 1 Identify how each transformation affects
the constant in vertex form.
Translation 2 units left h 2
Translation 5 units down k 5
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Example 1 continued
Step 2 Write the transformed function.
g(x) a(x h)2 k
Vertex form of a quadratic function
Simplify.
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Check Graph both functions on a graphing
calculator. Enter f as Y1, and g as Y2. The
graph indicates the identified transformations.
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Example 2
Use the description to write the quadratic
function in vertex form.
Step 1 Identify how each transformation affects
the constant in vertex form.
Translation 2 units right h 2
Translation 4 units down k 4
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Example 2 Continued
Step 2 Write the transformed function.
g(x) a(x h)2 k
Vertex form of a quadratic function
Simplify.
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Example 2 Continued
Check Graph both functions on a graphing
calculator. Enter f as Y1, and g as Y2. The
graph indicates the identified transformations.
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Example 3
Use the description to write the quadratic
function in vertex form.
The parent function f(x) x2 is reflected across
the x-axis and translated 5 units left and 1 unit
up to create g.
Step 1 Identify how each transformation affects
the constant in vertex form.
Reflected across the x-axis a is negative
Translation 5 units left h 5
Translation 1 unit up k 1
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Example 3 Continued
Step 2 Write the transformed function.
g(x) a(x h)2 k
Vertex form of a quadratic function
(x (5)2 (1)
Substitute 1 for a, 5 for h, and 1 for k.
Simplify.
(x 5)2 1
g(x) (x 5)2 1
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Example 3 Continued
Check Graph both functions on a graphing
calculator. Enter f as Y1, and g as Y2. The
graph indicates the identified transformations.
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Example 4 Scientific Application
On Earth, the distance d in meters that a dropped
object falls in t seconds is approximated by
d(t) 4.9t2. On the moon, the corresponding
function is dm(t) 0.8t2. What kind of
transformation describes this change from d(t)
4.9t2, and what does the transformation mean?
Examine both functions in vertex form.
dm(t) 0.8(t 0)2 0
d(t) 4.9(t 0)2 0
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Example 4 Continued
The value of a has decreased from 4.9 to 0.8. The
decrease indicates a vertical compression. Find
the compression factor by comparing the new
a-value to the old a-value.
The function dm represents a vertical compression
of d by a factor of approximately 0.16. Because
the value of each function approximates the time
it takes an object to fall, an object dropped
from the moon falls about 0.16 times as fast as
an object dropped on Earth.
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Example 5 Continued
Check Graph both functions on a graphing
calculator. The graph of dm appears to be
vertically compressed compared with the graph of
d.

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Example 5
The minimum braking distance d in feet for a
vehicle on dry concrete is approximated by the
function (v) 0.045v2, where v is the vehicles
speed in miles per hour.
The minimum braking distance dn in feet for
avehicle with new tires at optimal inflation is
dn(v) 0.039v2, where v is the vehicles speed
in miles per hour. What kind of transformation
describes this change from d(v) 0.045v2, and
what does this transformation mean?
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Example 5 Continued
Examine both functions in vertex form.
d(v) 0.045(t 0)2 0
dn(t) 0.039(t 0)2 0
The value of a has decreased from 0.045 to 0.039.
The decrease indicates a vertical
compression. Find the compression factor by
comparing the new a-value to the old a-value.
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Example 5 Continued
Check Graph both functions on a graphing
calculator. The graph of dn appears to be
vertically compressed compared with the graph of
d.