1. 3.5-3 Symmetry, Monotonicity

2. In some cases, we will have some kind of symmetry in regards to graphs of a function – Symmetry = a specific part or portion of a graph is reflected (mirrored) To identify symmetry, we can – A) Study the graph of a function – B) Use specific test values

3. Origin A graph is symmetric to the origin if: – f(-x) = -f(x) – Plugging in opposite x-values yields opposite y- values – Called “odd” functions Example. Is the function f(x) = -2x 3 + 8x symmetric to the origin?

4. Y-axis A graph is symmetric to the y-axis if f(-x) = f(x) – Opposite x-values yields the same y-values – Called “even” functions Example. Is the graph f(x) = 2x 2 symmetric to the y-axis?

5. Distinction Functions vs. Equations Remember, a function is written as f(x) = …, whereas an equation may just be y = x. An equation is symmetric to: – Y-axis: if replacing x with –x gives the same results – X-axis: if replacing y with –y gives the same results – Origin: if replace x with –x and y with –y gives the same results

6. X-axis A graph is symmetric to the x-axis if f(x) = -f(x) – Replace y with –y – Problem with this? Example. Is the relation x = y 2 symmetric to the x-axis?

7. Determine if the following functions are odd, even, neither. Determine if the equation is symmetrical to the y-axis, x-axis, or origin. A) f(x) = |x| + 3 B) x = -y 2 C) m(x) = √x - 1

8. Increasing, Decreasing A function is considered increasing on the interval x 1 to x 2 if f(x 1 ) > f(x 2 ) A function is considered decreasing on the interval x 1 to x 2 if f(x 1 ) < f(x 2 ) A function is considered constant on the interval x 1 to x 2 if f(x 1 ) = f(x 2 )

9. To determine increasing vs. decreasing, best done using a graph, or at least a visualization

10. Example. Find the intervals of increasing/decreasing for the function f(x) = -|x - 2| – Graph?

11. Example. The rate of cost incurred by a newspaper stand is given by: P(x) = -3√x + 4, x ≥ 0, x < 3 -x + 8, x ≥ 3 Determine the intervals of increasing or decreasing.

12. Assignment Page. 257 35-55 odd, 60-63