In the previous section, you used a generic function to examine patterns in horizontal and vertical translations of functions. At the same time, you should have noticed the effect of the geometric patterns upon the parameters in the algebraic representation of the functions. In this section, you will use similar reasoning to examine how vertical stretches or compressions affect the algebraic representations of the functions.

In the interactive below, stretch or compress the function vertically by using the slider. As you do so, notice how the coordinates change in comparison to the scale factor being used to stretch or compress the function. Use the interactive to help you answer the questions that follow.

Interactive popup. Assistance may be required.

Need additional directions?

• Create a table of values for x, f(x), and 2.5f(x).

  x	f(x)	2.5f(x)

  
  
  Interactive popup. Assistance may be required.

  Check Your Answer

  x	f(x)	2.5f(x)
  -2	1	2.5
  -1	-1	-2.5
  0	2	5
  1	1	2.5
  2	3	7.5
  3	0	0

  
• How does the change in the graph of the function compare to the equation, y = 2.5f(x)?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The graph is a vertical stretch by a factor of 2.5, which matches the coefficient of f(x).

• Create a table of values for x, f(x), and 0.25f(x).

  x	f(x)	0.25f(x)

  
  
  Interactive popup. Assistance may be required.

  Check Your Answer

  x	f(x)	0.25f(x)
  -2	1	0.25
  -1	-1	-0.25
  0	2	-0.50
  1	1	0.25
  2	3	0.75
  3	0	0

  
• How does the change in the graph of the function compare to the equation, y = 0.25f(x)?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The graph is a vertical compression by a factor of 0.25, which matches the coefficient of f(x).

Pause and Reflect

Why does a coefficient of f(x) between 0 and 1 generate a vertical compression while a coefficient of f(x) greater than 1 generates a vertical stretch?

Interactive popup. Assistance may be required.

Check Your Answer

The coefficient of f(x) multiplies the range values of f(x) by the same number. If that number is between 0 and 1, then the resulting y-values will be closer to the x-axis, and the graph will appear to be compressed. If that number is greater than 1, then the resulting y-values will be farther from the x-axis, and the graph will appear to be stretched.

Practice

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

   If the graph is vertically stretched by a factor of 4, what will be the equation of the new function?

   Interactive popup. Assistance may be required.

   Need a hint?

   The general form y = a(x – h)² + k can be used to represent transformed quadratic functions. Which parameter controls a vertical stretch?
   Interactive popup. Assistance may be required.

   Check Your Answer

   y = 4x²
2. The graph of the quadratic function f(x) is shown below.
   

   If the graph is vertically compressed by a factor of 1 over 8 1 8 , what will be the equation of the new function?

   Interactive popup. Assistance may be required.

   Need a hint?

   The general form y = a(x – h)² + k can be used to represent transformed quadratic functions. Which parameter controls a vertical compression? Use the coordinates of the vertex to help you write the equation of the original function.
   Interactive popup. Assistance may be required.

   Check Your Answer

   y = 1 over 8 - 1 8 (x + 3)² + 6
3. Four exponential functions are shown below.

   f(x) = 5(2)^x           g(x) = 2.4(2)^x         h(x) = 1 over 7 1 7 (2)^x          j(x) = 0.75(2)^x

   Write the functions in order from the least to the greatest vertical stretch.

   Interactive popup. Assistance may be required.

   Need a hint?

   The general form y = a(2)^{x– h} + k can be used to represent transformed exponential function with a base of 2. Which parameter controls vertical stretches or compressions? List those parameters in order from least to greatest.
   Interactive popup. Assistance may be required.

   Check Your Answer

   h(x) = 1 over 7 1 7 (2)^x,  j(x) = 0.75(2)^x, g(x) = 2.4(2)^x, f(x) = 5(2)^x