4 4 periodic functions; stretching and translating
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This document discusses periodic functions and techniques for stretching, shrinking, and translating graphs of functions. It defines a periodic function as one where f(x+p)=f(x) for some period p and all x in the domain. The smallest such p is the fundamental period. Examples are given of finding the period and amplitude of functions from graphs, as well as stretching and shrinking graphs horizontally and vertically by multiplying or dividing the x-coordinates, and translating graphs by adding numbers to x or y.
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4 4 periodic functions; stretching and translating
- 1. 4-4 PERIODIC FUNCTIONS; STRETCHING AND TRANSLATING GRAPHS Objectives: 1. Determine period and amplitude from graphs 2. Stretch and shrink graphs horizontally and vertically. 3. Translate graphs.
- 2. PERIODIC FUNCTIONS A function is periodic if there is a positive # p, called the period, such that: f(x + p) = f(x) for all x in the domain of f. The smallest period is the fundamental period of the function. So, if f is periodic, then f(x) = f(x + mp) for all x and any integer m.
- 3. EXAMPLE 1 Find the fundamental period of f. Find f(99)
- 4. AMPLITUDE Amplitude = max – min 2 Example 2: Find the amplitude of:
- 5. YOU TRY! Find the fundamental period. Find f(45). Find the amplitude.
- 6. STRETCHING AND SHRINKING y = cf(x) is a vertical stretch/shrink c > 1: stretch 0 < c < 1: shrink x-coordinates don’t change Multiply y-coordinates by c
- 7. y = f(cx) is a horizontal stretch/shrink c > 1: shrink 0 < c < 1: stretch Divide x-coordinates by c y-coordinates don’t change
- 8. YOU TRY! Graph y = 2f(x) Graph y = f(2x)
- 9. TRANSLATING (SHIFTING) Numbers added to x: Shift graph horizontally Need to “undo” so shift opposite direction Numbers added to y (not in () with x): Shift graph vertically
- 10. EXAMPLE 3 Sketch: y = |x| y – 2 = |x – 3| y = |x + 5|