[4.3] Reflecting Graphs, & Symmetry

1. [4.3] Reflecting Graphs, & Symmetry Objective: To reflect graphs and use symmetry to sketch graphs.

2. 1) The graph y = – f(x) is found by reflecting y = f(x) over the x-axis. (2, 4) Reflection over x-axis: 2) All coordinates (x,y) on f(x) become (x, – y) on –f(x) 3) The opposite output, results in a vertical flip, over a horizontal axis. (2, –4)

3. Reflection over y-axis: 1) The graph y = f(– x) is found by reflecting y = f(x) over the y-axis. 2) All coordinates (x,y) on f(x) become (– x, y) on f(– x) 3) The opposite input, results in a horizontal flip, over a vertical axis. (–4, 2) (4, 2)

4. Reflection over the line y = x: (1, 3) 1) The reflection of the graph of f(x) over the line y = x is found by interchanging the x and y variables (3, 1) Sound familiar? Inverse! 2) All coordinates (x,y) on f(x) become (y, x) on the reflection

5. Absolute Value of a Function: 1) | f(x)| is found by taking all the negative y-values and making them positive. 2) Reflect all the coordinates with negative y-values over the x-axis.

6. These tests are designed to tell you about the symmetry of a graph without using a visual graph. We may use some functions you are familiar with to begin the process, but ultimately you should be using algebra to determine symmetry. Algebraic Tests for Symmetry:

7. 1) In the equation, replace y with –y. 2) If the resulting equation is the same as the original equation, then there is symmetry over the x-axis. Test for Symmetry about the x-axis: Ex: Replace the y : Compare new and original. Are they equivalent? YES they are equivalent, so the function is symmetric over x-axis

8. 1) In the equation, replace x with –x. 2) If the resulting equation is the same as the original equation, then there is symmetry over the y-axis. Test for Symmetry about the y-axis: Ex: Replace the x : Compare new and original. Are they equivalent? NO they are not equivalent, so the function is not symmetric over y-axis

9. 1) In the equation, interchange all of the x’ s and y’ s 2) If the resulting equation is the same as the original equation, then there is symmetry over the line y = x. Test for Symmetry Over the line y = x : Ex: Interchange the variables: Compare new and original. Are they equivalent? YES the function is symmetric over the line y = x.

10. 1) In the equation, replace x with –x, AND replace y with –y. 2) If the resulting equation is the same as the original equation, then there is symmetry about the origin. Test for Rotational Symmetry about Origin: Ex: Make replacements: Compare new and original. Are they equivalent?

11. Compare new and original. Are they equivalent? Watch: YES the function has rotational symmetry about the origin.