In this section, you will learn about vertically shrinking (compressing) and stretching exponential and logarithmic graphs.

Below are four functions, four graphs, and four descriptions of translations to the graphs of some exponential and logarithmic functions. Some functions, graphs, and descriptions have already been placed in the table. Place the remaining functions, graphs, and descriptions in the correct place.

Conclusion Questions

• Which parameter seems to generate a vertical compression or stretch?

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  The coefficient of the parent function, a, generates a vertical compression or stretch.

• How can you tell whether the parameter will generate a vertical compression or a vertical stretch?

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  If |a| > 1, then the parameter will generate a vertical stretch. If 0 < |a| < 1, then the parameter will generate a vertical compression.

• How can you tell whether the parameter will generate a reflection across a horizontal line?

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  If the sign of the parameter, a, is negative, then it will generate a reflection of the parent function across a horizontal line.

Pause and Reflect

How does the vertical stretch, vertical compression, or vertical reflection of exponential and logarithmic functions compare to the same transformations of other functions, such as quadratic, square root, or rational functions?

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These vertical transformations work the same, regardless of the function family. The sign of the coefficient of the parent function, a, tells you whether or not there will be a reflection of the parent function across a horizontal line. The magnitude of the parameter, a, tells you whether there will be a vertical stretch or a vertical compression.

Practice

1. Describe how the -6 in the following equation changes the graph of the parent function, p(x) = logx.
   r(x) = −6log(x − 9) − 3

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   Which transformation does the coefficient of the parent function affect?

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   The graph of p(x) is reflected across the horizontal line and vertically stretched by a factor of 6.

2. If the graph of a function is vertically compressed by a factor of 1 over 5 1 5 from the parent function, how is the equation of the parent function changed?

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   Which parameter controls a vertical compression?

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   The equation of the parent function has been multiplied by 1 over 5 1 5 .

3. What has happened to the equation k(x) = 10^x, to generate j(x), the function whose graph is shown below?
   

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   What would be the y-intercept of the graph of k(x)? Use that point to determine the transformation.

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   k(x) = 10^x has been changed to j(x) = -1(10)^x.