Periodic Functions, Stretching, and Translating Functions Section4-4 Periodic Functions, Stretching, and Translating Functions Objectives: To determine periodicity and amplitude from graphs To stretch and shrink graphs both vertically and horizontally, and to translate graphs.

Periodic Functions What does it mean to be periodic? 1. Having or marked by repeated cycles. 2. Happening or appearing at regular intervals. 3. Recurring or reappearing from time to time; intermittent. What types of things are periodic? Tides (they rise and fall daily) Amount of daylight per day (increases and decreases within a period of a year) The amount of the moon that is visible (moon phases).

Periodic Functions periodic Definition: A function f is periodic if there is a positive number p, called a period of f, such that f(x + p) = f(x) f(x + p) 4 Example: periodic f(-4 + 4) = f(0) = 1.5 f(-4) = 1.5 f(x) f(-2) = -1.5 f(6) = -1.5 8 Definition: The fundamental period is the smallest period of a periodic function. Here the fundamental period is 4. Note: if you start at the origin, it takes 4 units to get back to where you were starting. Question: What is f(99)? Since f has a fundamental period of 4, perform 99/4 = 24. This tells us the function repeats itself 24 times. F(99) = f(99 - 24*4) = f(3) = 0

Periodic Functions Definition: If a periodic function has a maximum value M and a minimum value m, then the amplitude A of the function is given by: A = M - m 2 Find the amplitude from the previous example. M = 1.5 m = -1.5 A = 1.5 – (-1.5) 2 A = 1.5 +1.5 2 A = 3 = 1.5 2

Stretching and Shrinking Graphs Previously we saw a relationship between the equation of a function and the symmetry of its graph. Now see a relationship between the equation of a function and how much the graph will shrink or stretch. The graph of y = cf(x) where c is positive (and not equal to one) is a vertical stretch or vertical shrink of y = f(x) Example: vertical shrink Example: vertical stretch y = 2f(x) y = 1/2f(x) y = f(x) y = f(x) 3 -1 P = 4 P = 4 A = 1 A = 1 P = 4 P = 4 A = 2 A = 1/2

Changing the Period and Amplitude The graph of y = f(cx) where c is positive (and not equal to one) is a horizontal stretch or horizontal shrink of y = f(x) Example: horizontal shrink Changing the Period and Amplitude of a Periodic Function If a periodic function f has period p and amplitude A, then: y = cf(x) has period p and amplitude cA, and y = f(cx) has period p/c and amplitude A. y = f(2x) y = f(x) 3 -1 P = 4 A = 1 P = 2 A = 1 Example: horizontal stretch y = 1/2f(x) y = f(x) P = 4 A = 1 P = 8 A = 1

Translating Graphs In order to move a graph horizontally (left or right) we must use a variation of the original function Horizontal Shift to the left 2 units f(x + 2) Horizontal Shift to the right 2 units f(x - 2) Original function f(x) Y = (x + 2)2 Y = (x - 2)2 Y = X2 Shift f(x) vertically k units Shift f(x) horizontally h units Note: y = f(x – h) + k

Translating Graphs In order to move the graph vertically (up or down) we must use a variation of the original function. Vertical Shift up 2 units f(x) + 2 Vertical Shift down 2 units f(x) - 2 Original function f(x) Y = x2 + 2 Y = x2 - 2 Y = x2 Shift f(x) vertically k units Shift f(x) horizontally h units Note: y = f(x – h) + k

Translating Graphs If the equation y = f(x) is changed to: Then the graph of y = f(x) is: Reflected in the x-axis Unchanged when f(x) ≥ 0 and reflected in the y-axis Reflected in the y-axis Reflected in the line y = x Stretched vertically Shrunk vertically Shrunk horizontally Stretched horizontally Translated h units horizontally and k units vertically

Section 4-5 Inverse Functions Objective: To find the inverse of a function, if the inverse exists.

Homework p143-144: 1-4 (all), 5-15 (odd)