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Lesson 8.2 Translating Graphs To write equations to describe translations of the absolute-value and squaring functions. To graph and recognize translations of the absolute value and squaring functions. To explore the concept of a family of functions.

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In previous chapters you wrote many linear and exponential functions. You used points, y-intercepts, slope, starting values, and constant multipliers to write equations “from scratch.” In this chapter you will use transformations to base functions such as y = |x| and y = x 2.

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Translations of Functions First you’ll transform the absolute-value function by making changes to x. Enter y = |x| into y1 and graph it on your calculator. If you replace x with x-3 in the function y = |x|, you get y = |x-3|. Enter y2=|x-3| and graph it. How have you transformed the graph of y = |x|? 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|. How did these two points help verify the transformation you just performed?

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Find a function for y2 that will translate the graph of left 4 units. What is the function? In the equation y = |x|, what did you replace x with to get your new function?

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Write a function for y2 to create each graph below. Check your work by graphing both y1 and y2.

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Next you will transform the absolute-value function by making changes to y. Clear all of the functions. Enter y1= |x| and graph it. If you replace y with y-3 in the function y= |x|, you get y- 3=|x|. Solve it for y and you get y=|x|+3. Enter y2=|x|+3. Graph it. 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|? Name the coordinates of the vertex of both graphs. How do these two points help verify the transformation you just found.

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Find a function y2 that will translate the graph of y=|x| down 3 units. What is the function? In the function, y=|x|, what did you replace y with to get your new function?

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Write a function for y2 to create each graph below. Check your work by graphing both y1 and y2.

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Summarize what you have learned about translating the absolute-value graph vertically and horizontally.

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Definitions The most basic form of a function is called a parent function. By transforming the graph of a parent function, you can create infinitely many new functions or a family of functions. y=|x-3| and y=|x| +3 are members of the absolute-value family of functions.

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Example A The graph of the parent function y=x 2 is shown in bold. Its image after a transformation is shown in a thin line. Study the transformation and write the equation for the transformed graph.

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Example B The starting number of bacteria in a culture dish is unknown, but the number grows by approximately 30% each hour. After 4 hours, there are 94 bacteria present. Write an equation to model this situation. Then find the starting number of bacteria.

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The starting number is unknown, but you can find it by assuming that your beginning with 94 bacteria, and then shifting back in time. If you began with 94 bacteria the function would be y=94(1+0.30) x, where x represents the time elapsed in hours y represents the number of bacteria The black graph represents this function. However, there were 94 bacteria after 4 hrs, not at 0 hrs. So translate the point (0,94) to (4,94). This changes the equation to y=94(1+0.30) x-4

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