In this section, you will investigate how translating the graph of a parent function, both horizontally and vertically, affects its algebraic representation. Once you do so, then you will use parameters in equations to make predictions about how the graph of the parent function would change.

Let’s start out by investigating how the horizontal translation and vertical translation of a generic function changes the coordinates of the function.

In the interactive below, translate the function by using the sliders. As you do so, notice how the coordinates change in comparison to the direction and distance of the translation. Use the interactive to help you answer the questions that follow.

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Vertical Translations

• Create a table of values for x, f(x), and f(x) + 2.

x	f(x)	f(x) + 2



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Check Your Answer
x	f(x)	f(x) + 2
-2	1	3
-1	-1	1
0	2	4
1	1	3
2	3	5
3	0	2

• How does the change in the graph of the function compare to the equation, y = f(x) + 2?

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  Check Your Answer

  The graph is a vertical translation upward 2 units, which matches the constant term + 2.

• Create a table of values for x, f(x), and f(x) – 4.

x	f(x)	f(x) – 4



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x	f(x)	f(x) – 4
-2	1	-3
-1	-1	-5
0	2	-2
1	1	-3
2	3	-1
3	0	-4



• How does the change in the graph of the function compare to the equation, y = f(x) – 4?

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  Check Your Answer

  The graph is a vertical translation downward 4 units, which matches the constant term −4.

Horizontal Translations

• Create a table of values for x, x – 3, and f(x – 3) using the given x-values.

  x	x – 3	f(x – 3)
  1
  2
  3
  4
  5
  6

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  Check Your Answer

  x	x – 3	f(x – 3)
  1	-2	1
  2	-1	-1
  3	0	2
  4	1	1
  5	2	3
  6	3	0

  

• How does the change in the graph of the function compare to the equation, y = f(x – 3)?

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  Check Your Answer

  The graph is a horizontal translation 3 units to the right, which is the same distance but the opposite direction as the addend to x.

• Create a table of values for x, x + 2, and f(x + 2) using the given x-values.

  x	x + 2	f(x + 2)
  -4
  -3
  -2
  -1
  0
  1

  
  

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  Check Your Answer

  x	x + 2	f(x + 2)
  -4	-2	1
  -3	-1	-1
  -2	0	2
  -1	1	1
  0	2	3
  1	3	0

  

• How does the change in the graph of the function compare to the equation, y = f(x + 2)?

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  Check Your Answer

  The graph is a horizontal translation 2 units to the left, which is the same distance but the opposite direction as the addend to x.

Pause and Reflect

When translating a function, why does a vertical translation match the sign and distance for the constant term, but the horizontal translation is the same distance but the opposite sign?

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Check Your Answer

Vertical translations are performed by adding (or subtracting) a number after the parent function is performed to the x-value. Horizontal translations occur before using the parent function by adding or subtracting a number from the x-value.

Practice

1. The graph of the parent quadratic function, y = x², is shown below.
   

   If the graph is translated 4 units to the right and 3 units down, what will be the equation of the new function?

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   The general form y = a(x – h)² + k can be used to represent transformed quadratic functions. Which two parameters control horizontal and vertical translations?
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   Check Your Answer

   y = (x – 4)² – 3
2. The graph of the quadratic function f(x) is shown below.
   

   If the graph is translated 6 units to the right and 2 units up, what will be the equation of the new function?

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   The general form y = a(x – h)² + k can be used to represent transformed quadratic functions. Which two parameters control horizontal and vertical translations? Use the coordinates of the vertex to help you.
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   Check Your Answer

   y = -(x – 3)² + 8
3. The graph of the square root parent function, f(x) = √x, is shown below.
   

   If the parent function is translated 7 units to the left and 4 units up, what will be the equation of the new function?

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   The general form y = a√x – h + k can be used to represent transformed square root functions. Which two parameters control horizontal and vertical translations? Use the coordinates of the endpoint to help you.
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   Check Your Answer

   y = √x + 7 + 4