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Reflections and the Square Root Curve

Reflections and the Square Root Curve. Lesson 4.5. The graph of the square root function, , is another parent function that you can use to illustrate transformations. From the graphs below, what are the domain and range of ?.

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Reflections and the Square Root Curve

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  1. Reflections and the Square Root Curve Lesson 4.5

  2. The graph of the square root function, , is another parent function that you can use to illustrate transformations. From the graphs below, what are the domain and range of ? Graph on your calculator. You can see that at x=3, f(x)≈ 1.732.

  3. The graph of the square root function, , is another parent function that you can use to illustrate transformations. From the graphs below, what are the domain and range of ? What is the approximate value of at x = 8? What happens when you try to find f (x) for values of x < 0? How would you use the graph to find ?

  4. Take a Moment to Reflect • In this investigation you first will work with linear functions to discover how to create a new transformation—a reflection. Then you will apply reflections to quadratic functions and square root functions.

  5. Predict what the graph of -f1(x) will look like. Then check your prediction by graphing f2(x)= -f1(x). • b. Change f1 to f1(x)=-2x-4, and repeat the instructions in part a. • c. Change f1 to f1(x)=x2 +1 and repeat. • d. In general, how are the graphs of y=f(x) and y=-f (x) related? Graph on your calculator.

  6. Predict what the graph of f1(-x) will look like. Then check your prediction by graphing f2(x)= f1(-x). • b. Change f1 to f1(x)=-2x-4, and repeat the instructions in part a. • c. Change f1 to f1(x)=x2 +1 and repeat. • Change f1 to f1(x)=(x-3)2+2 and repeat. Explain what happens. • In general, how are the graphs of y=f(x) and y=f (-x) related? Graph on your calculator.

  7. Graph on your calculator. Predict what the graphs of f2 =- f1(x) and f3= f1(-x) will look like. Use your calculator to verify your predictions. Write equations for both of these functions in terms of x. Predict what the graph of f4 =-f1(-x) will look like. Use your calculator to verify your prediction. Notice that the graph of the square root function looks like half of a parabola. Why isn’t it an entire parabola? What function would you graph to complete the bottom half of the parabola?

  8. Reflection of a Function • A reflection is a transformation that flips a graph across a line, creating a mirror image. • Given the graph of y=f(x), • the graph of y=f(-x) is a horizontal reflection across the y-axis, and • the graph of -y= f(x), or y=-f(x), is a vertical reflection across the x-axis.

  9. Example A piecewise function is a function that consists of two or more ordinary functions defined on different domains. Graph

  10. Enter the equation in your graphing calculator.

  11. Find an equation for the piecewise function pictured at right. For -4≤x≤0, the function appears to be equal to a reflection of the square root function about the y-axis and then shifted one unit up. This would be the function For 0<x≤3, the function appears to be equal to a reflection of the square curve reflected over the x-axis and then shifted one unit to the right and 2 units up.

  12. Find an equation for the piecewise function pictured at right.

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