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Stretching and Shrinking Graphs

In Lesson 9.4 on page 501 of Discovering Algebra: An Investigative Approach, you see a picture of a butterfly stretching and shrinking vertically. The first sketch on this page allows you to transform that same picture. You can use the second sketch to stretch and shrink vertically the quadrilateral in the investigation that begins on page 502. The third sketch is for stretching and shrinking graphs of functions, as in the examples on pages 504–505.

Sketch

Drag the red point on the slider in the sketch below to see the butterfly stretch or shrink.

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Investigate

  1. What dimension does the slider change?
  2. How can you stretch the butterfly? How can you shrink it?

Sketch

The sketch below shows a quadrilateral determined by the points (1, 3), (2, –1),
(–3, 0), and (–2, 2). You can multiply the y-coordinates of these points by the value of a on the slider.

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Investigate

  1. Look at what happens to the quadrilateral when a has values 2, 3, 0.5, and –2.
  2. What happens to the coordinates of the point (–2, 2) when the stretch factor is 2? 3? 0.5? –2?
  3. Why doesn’t the point (3, 0) change when stretched?
  4. What happens when the stretch factor is 1?
  5. What happens when the stretch factor is greater than 1?
  6. What happens when the stretch factor is greater than 0 but less than 1?
  7. What happens when the stretch factor is 0?
  8. What happens when the stretch factor is negative?

Sketch

You can stretch or shrink the graph of a function vertically by changing the expression of the function. In the sketch below, you can choose to study a line, the absolute-value function, or a parabola. Stretch or shrink the function by changing the value of a using the slider. You can change the point on the parent function by dragging it.

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Investigate

  1. What values of a give a vertical shrink? What values give a vertical stretch?
  2. How can you tell from the graph of the function and its parent what value of a was used to stretch or shrink the function?
  3. To stretch or shrink a function graph vertically, the right side of the function’s equation is multiplied by a. What changes to the equation would stretch or shrink a function horizontally instead of vertically?