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Stretches and Shrinks of Shapes and Functions

On this web page you will investigate how graphs can be stretched or shrunk by changing their equations. This exploration will reinforce the concepts in Lesson 4.6 of Discovering Advanced Algebra: An Investigative Approach.

Sketch

This sketch shows a green polygon and its stretch or shrink by the scale factor 0.5, colored blue. You can change the scale factor by dragging point P.

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Investigate

  1. What happens when you change the value of the scale factor P?
  2. What happens when P equals zero? When P equals 1?
  3. What happens when P is greater than 1?

Sketch

This sketch shows the graph of the function y=a|(1/b)x|. Sliders allow you to change the values of a and b.

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Investigate

  1. What values of a give a vertical stretch? A vertical shrink?
  2. How does the graph differ if a is positive or negative? Equal to 0?
  3. How does the graph differ when a is between 0 and 1?
  4. Summarize your findings about the effect of coefficient a on the graph.
  5. What values of b give a horizontal stretch? A horizontal shrink?
  6. How does the graph differ if b is positive or negative? Equal to 0?
  7. How can you describe the graph when b is between 0 and 1?
  8. Summarize your findings about the effect of coefficient b on the graph.
  9. What happens when a is 2 and b is 3? Explain.
  10. If a is 3, can you find a value of b that results in the same graph as that of  y = |x|? Explain.

Sketch

This sketch shows the graph of the function  y=a(1/bx)^2+c. Sliders allow you to change the values of a, b, and c.

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Investigate

  1. What values of a give a vertical stretch? A vertical shrink?
  2. How does the graph differ if a is positive or negative? Equal to 0?
  3. How does the graph differ when a is between 0 and 1?
  4. Summarize your findings about the effect of coefficient a on the graph.
  5. What values of b give a horizontal stretch? A horizontal shrink?
  6. How does the graph differ if b is positive or negative? Equal to 0?
  7. How does the graph differ when b is between 0 and 1?
  8. Summarize your findings about the effect of coefficient b on the graph.
  9. What does the graph look like when a is 2 and b is 3? Explain.
  10. If a is 4, can you find a value of b that results in the same graph as that of y = x2? Explain.
  11. Now change c. What does the graph look like when c is positive? Negative? Equal to 0? Explain.
  12. Can you find values of a, b, and c that make the graph pass through all three points (0, 3), (2, 1), and (???4, ???5)? Describe your approach.

Sketch

This sketch shows the graph of the function  y=af(1/bx) + c and sliders that allow you to change the values of a, b, and c.

 

Sorry, this page requires a Java-compatible web browser.

Investigate

  1. What values of a give a vertical stretch? A vertical shrink?
  2. How does the graph differ if a is positive or negative? Equal to 0?
  3. How does the graph differ when a is between 0 and 1?
  4. Summarize your findings about the effect of coefficient a on the graph.
  5. What values of b give a horizontal stretch? A horizontal shrink?
  6. How does the graph differ if b is positive or negative? Equal to 0?
  7. How does the graph differ when b is between 0 and 1?
  8. Summarize your findings about the effect of coefficient b on the graph.
  9. What does the graph look like when a is 2 and b is 3? Explain.
  10. Now change c. What does the graph look like when c is positive? Negative? Equal to 0? Explain.
  11. Summarize what you have learned about stretching and shrinking graphs vertically and horizontally.