Name_
Class
Date_
= X2
Translating the Graph of f(x)
Essential question: How does the graph of g(x) = (x — h) 2 + k compare with the
graph off(x) = x2?
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ENGAGE \ Understanding the Parent Quadratic Function
A quadratic function is a function that can be written in the form f(x) = ax2 + bx + c
where a # 0. The most basic quadratic function isf(x) = x 2. This function is often called
the parent quadratic function. Once you understand the graph of the parent function,
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you can understand the graphs of other quadratic functions. To graph f(x) = x2, make a
table of values, plot the ordered pairs, and draw the graph.
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The graph of a quadratic function is called a parabola, which can either open up (as the ,
graph of the parent quadratic function does) or open down. The highest or lowest point
on a parabola is called the vertex. The axis of symmetry divides the parabola into two
congruent halves and passes through the vertex. For the graph off(x) = x 2, the vertex is
(0, 0) and the axis of symmetry is they-axis, or x = 0.
When the graph of a quadratic function opens up, the y-coordinate of the vertex is the
function's minimum value, or simply Minimum. When the graph of a quadratic function
opens down, the y-coordinate of the vertex is the function's maximum value, or simply
maximum. The function f(x) = x2 has a minimum of 0.
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REFLECT
1a.
What is the domain off(x) = x2? What is the range?
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For what values of x is f(x) = x2 increasing? For what values of x is it decreasing?
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Unit 2
31
Lesson 1
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iAMPLE\ Graphing Quadratic Functions
Graph each quadratic function. (The graph of the parent function is
shown.)
A
g(x) = x2 + 1
g(x) = x2 + 1
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REFLECT \
2a. Identify the vertex of the graph of g(x) = x2 + 1, and give the domain and range of
the function. Describe the graph of (x) = + I as a translation of the graph of the
parent function f(x) = x2.
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2b. Identify the vertex of the graph of g(x) = x2 -- 3, and give the domain and range of
the function. Describe the graph of g(x) = x2 r 3 as a translation of the graph of the
parent function f(x) = x 2 .
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Unit 2
32
Lesson
1
2c.
Describe the graph of g(x) = x2 + 5 as a translation of the graph of the parent
function. Identify the vertex of the graph of g(x) = x2 + 5, and give the domain and
range of the function.
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2d. Write a general statement describing how the graph of g(x) = x2 + k is related
to the graph of the parent function f(x) = x2. Identify the vertex of the graph of
g(x) = x2 + k, and give the domain and range of the function.
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EXAMPLE
Graphing Quadratic Functions
Graph each quadratic function. tThe graph of the parent function is shown.)
A
g(x) (x + 1) 2
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1
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1
1
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Lesson 1
REFLECT
3a. Identify the vertex of the graph of 8(x) = (x Jr 1)2 , and give the domain and range of
the function. Describe the graph of g(x) = (x + 1)2, axa translation of the graph of
the parent function f(x) = x2.
3b. Identify the vertex of the graph of g(x) = (x — 3)2 , and give the domain and range of
the function. Describe the graph of g(x) (x — 3)2 as a translation of the graph of.
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' The parent function 'f(x) = x2. •
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3c. Write a general statement describing how the graph of g(x) = (x — 102 is related to
the graph of the parent function f(x) -- x2 . Identify the vertex of the graph of
g(x) -= (x — h)2 , and give the domain and range of the function.
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EXAMPLE \ Writing Equations of Quadratic Functions
Write the equation of the quadratic function whose graph is shown.
.A
Compare the given graph to the graph of the parent function f(x) = x2.
Number of
Units
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B Determine the values of h and k for the function g(x) (x — h) 2 + k.
Ihl is the number of units the graph of the parent function is translated horizontally.
For a translation to the f
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• Iki is the number of units the graph of the parent function is translated vertically.
For a translation up, k is
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Type of
Translation
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Complete the table to describe how the graph of the parent
function must be translated to get the graph shown.
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you check that the equation you wrote is correct?
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The graph of a quadratic function is a translation of the graph of the parent
function. Flow can you use the vertex of the translated graph to determine the
equation of the function?
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Graph the quadratic function.
1. g(x) = x2 + 3
2. g(x) = x2 — 4
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6. g(x) = (2 + x) 2
5. g(x) = (x + 1) 2
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8. g(x) = .(x + 11)+ 4
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Unit 2
35
Lesson 1
Write the equation of the quadratic function whose graph is shown.
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13. One function is given by g(4= (x —7)2 + 6.
Another function is graphed. Which function
has the greater minimum?
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14. One function is given by g(x)= (x + 5)2 - 3.
Another function is graphed. Which function
has the greater minimum?
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Identify the vertex of the graph of the function. Give the domain and range.
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18. f(x)=(x— 10) 2
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16. f(x) = x2 + 9
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19. .f(x) =(12 + x)2
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17. f(x) = x2 — 17
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20. f(x) = (x + 3)2
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21. Suppose you translate the graph off(x) .= (x— 4)2 + 3 left 4 units and down 3 units.
What is the equation of the resulting graph?
Unit 2
36
Lesson 1
C H ou ghton Miffli n H arcourt P ublishi n g Com pany
15. f(x) = x2