Dynamic Algebra™ Exploration

Translating Graphs

On this web page, you can use what you know about translating points to translate functions. This exploration will help you complete the investigation and Example B in Lesson 8.2 of Discovering Algebra: An Investigative Approach (pages 444–445 and 448).

Sketch

The sketch below shows the graphs of the absolute-value functions

y = |x|     and     y = |x − H| + K

You can change the numbers in the second function with the sliders—point H changes the number subtracted from x, and point K changes the number added to the entire function.

Investigate

1. Adjust the sliders so that you have the graph of y = |x − 3| + 0. This is the same as replacing x with x − 3. Think of the graph of y = |x| as the original figure and the graph of y = |x − 3| as its image. How have you transformed the graph of y = |x|?
2. The vertex of an absolute-value graph is the point where the function changes from decreasing to increasing or from increasing to decreasing. Name the coordinates of the vertex of the graph of y = |x|. Name the coordinates of the vertex of the graph of y = |x − 3|. Do these two points help verify the transformation you found in Question 1?
3. How can you translate the graph of y = |x| left 4 units? What is the function? In the equation y = |x|, what did you replace x with to get your new function?
4. Write functions for the image of y = |x| in each graph below. Check your work by using the sketch.
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5. Adjust the sliders in the sketch so that you have the graph of y = |x − 0| + 3. This is the same as replacing y with y − 3 and solving for y. Think of the graph of y = |x| as the original figure and the graph of y = |x| + 3 as its image. How have you transformed the graph of y = |x|?
6. Name the coordinates of the vertex of the graph of y = |x|. Name the coordinates of the vertex of the graph y = |x| + 3. Do these two points help you verify the transformation you found in Question 5?
7. Find the function that will translate the graph of y = |x| down 3 units. What is the function? In the function of y = |x|, what did you replace y with to get your new function?
8. Write functions for the image of y = |x| in each graph below. Check your work by using the sketch.
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Sketch

This sketch shows the graphs from Example B on page 448 of Discovering Algebra: An Investigative Approach. The red curve shows the number of bacteria as a function of time. The blue curve originally is the graph of the function y = 94(1 + 0.30)^x. You can translate the blue curve by dragging the red point.

Investigate

1. Drag the blue curve so that it lies on the red curve. Did you drag the point horizontally or vertically? How have you transformed the function?
2. The scale on the x-axis is 1. Explain why the equation of the red curve must be y = 94(1 + 0.30)^{(x − 4)}.
3. The blue curve goes through the point (0, 94). After 4 hours, the bacteria population is 94. What point represents this on the red curve? Does this point satisfy the equation of the red curve?
4. Use this equation to find the starting number of bacteria.