In the previous section, you used a generic function to examine patterns in the horizontal and vertical translations of functions. At the same time, you should have noticed the effect of those geometric patterns on the parameters in the algebraic representions of those functions. You also investigated vertical compressions and stretches of functions, and how parameters in equations affect those changes.

In this section, you will investigate how vertical reflections of functions affect the equations of those functions.

Four functions, four graphs and four descriptions of transformations are listed below. Some have already been placed in the table. Place the remaining objects next to their corresponding counterparts.

• Consider the general form of a parent function, y = f(x). How does the sign of the parameter, a, affect the graph of y = a ⋅ f(x)?

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  The sign of the parameter, a, can create a reflection across a horizontal axis. In this case, if a < 0, then the graph of the function is reflected across the x-axis.

Pause and Reflect

If a function has been vertically translated, how does the line of reflection relate to the vertical translation?

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The line of reflection is also vertically translated. For parent functions, the line of reflection would be the x-axis, or the line y = 0. When the function is vertically translated, the line of reflection moves the same number of units as the points contained in the graph of the function.

Practice

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

   If the graph was reflected across the line y = 0, what would 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 parameter controls a reflection across a horizontal line?
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   y = -x²
2. The graph of the quadratic function f(x) is shown below.
   

   If the graph were reflected across the line y = 6, what would 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 parameter will cause a reflection across a horizontal line? Use the coordinates of the vertex to help you write the equation of the original function.
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   y = (x + 3)² + 6
3. The graph of the function f(x) = 5 over x 5 x + 2 is shown in the graph below.
   

   If this graph were reflected across the line y = 2, what would be the equation of the new function?

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   The general form y = 22 over 7 a x – h + k can be used to represent transformed rational functions. Which parameter controls a reflection across a horizontal line?
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   g(x) = 5 over x -5 x + 2