1. Chapter 3: Transformations of Graphs and Data Lesson 4: Symmetries of Graphs Mrs. Parziale

2. Vocabulary Power function: is a function f with an equation in the form, where n is a positive integer greater than equal to 2. Even function: a function whose graph is symmetric with respect to the y-axis Odd function: a function whose graph is symmetric with respect to the origin.

3. Y-Axis Symmetry: Show with an instance that y=x 2 is symmetric to the y-axis INSTANCE OF SYMMETRY: (2,4) is on the graph f(2) = 2 2 = 4 (-2,4) is also on the graph f(-2) = (-2) 2 = 4

4. Three types of Symmetries (x-axis, y-axis, origin) y-axis: If (x,y) is on the graph, then the point (-x, y) is also on the graph. (reflective) x-axis: If (x,y) is on the graph, then the point (x, -y) is also on the graph. (reflective) origin: If (x,y) is on the graph, then the point (-x, -y) is also on the graph. (rotational)

5. Example 2: PROVE (for every case) that y=x 2 is symmetric to the y-axis Assume that (x,y) is on the graph then y = x 2 From the definition of quadratic functions (-x) 2 = x 2 By substitutiony = (-x) 2 Therefore (-x,y) is on the graph y = x 2 is symmetric to the y-axis

6. X-Axis Symmetry: Show with an instance that x=|y| is symmetric to the x-axis (2, 2) is on the graph f(2) = |2| = 2 (2, -2) is also on the graph f(-2) = |-2|= 2

7. Example 3: Prove that is symmetric to the x-axis Assume that (x,y) is on the graph Then x = |y| By definition of absolute value |-y| = |y| By substitutionx = |-y| Therefore (x, -y) is on the graph so it is symmetric to the x-axis

8. Origin Symmetry: Show with an instance that y = x 3 is symmetric to the origin. INSTANCE OF SYMMETRY: (1,1) is on the graph f(1) = 1 3 = 1 (-1, -1) is also on the graph f(-1) = (-1) 3 = -1

9. Example 4: Prove that y = x 3 is symmetric about the origin. Assume that (x,y) is on the graph Then y = x 3 By definition of cubic functions since y = x 3 then -y = -(x) 3 and (-x) 3 = -(x 3 ) By the transitive property of equality -y = (-x) 3 Therefore (-x,-y) is on the graph y = x 3 is symmetric to the origin

10. Example 5: Prove that the graph of is symmetric to the origin. Assume that (x,y) is on the graph Then y = 2x 5 +x 3 +x Multiply both sides by -1 -y = -(2x 5 +x 3 +x) Use distributive property-y = -2x 5 +-x 3 +-x -y = 2(-x) 5 +(-x) 3 +(-x) Therefore (-x,-y) is on the graph y = 2x 5 +x 3 +x is symmetric to the origin

11. Power functions and Even-Odd functions: Power functions are functions that have the form y = x n where n is an integer and. Even and Odd Functions: Even: f(-x) = ______________ Odd: f(-x) = ______________ f(x) -f(x)

12. Even Functions Even functions are symmetric to the y-axis Examples of even functions: y = |x| y = x 2 y =

13. Example 6: Prove the squaring function is an even function f(x) = x 2 For all x, (-x) 2 = x 2 Therefore, f(-x) = (-x) 2 = x 2 = f(x) Example: f(2) = (2) 2 = 4 f(-2) = (-2) 2 = 4

14. Odd Functions Odd functions are symmetric about the origin Examples of odd functions: y = x y = x 3 y =

15. Example 7: Prove f(x) = x 3 is an odd function

16. Note! **Not every function with the highest exponent as even is an even function. Example 8: (a) Show by counterexample that f(x) = x 4 + x is not an even function (b) Prove that f(x) = x 4 + x is not an even function.

17. Closure: Example 9: Prove or disprove: f(x) = x 3 is an even function.