VERTICAL STRETCHES AND COMPRESSIONS 6.3 VERTICAL STRETCHES AND COMPRESSIONS

Vertical Stretch: An Amplifier An amplifier takes a weak signal from a recording and transforms it into a stronger signal to power a set of speakers. Notice that the wave crests of the amplified signal are 3 times as high as those of the original signal; similarly, the amplified wave troughs are 3 times deeper than the original wave troughs. Amplified signal V(t) The figure shows a graph of an audio signal (in volts) as a function of time, t, both before and after amplification. Signal strength (volts) Original signal f(t) t, time

Example: Vertical Compression

Negative Stretch Factor The Figure gives a graph of a function y = f(x), together with a graph of y = − 2 · f(x) and the intermediate transformation y = 2 · f(x). The stretch factor is k = −2. First, the graph is stretched by a factor of 2, then it is reflected across the x-axis. Final y = -2 f(x) Original y = f(x) Intermediate y = 2 f(x)

Formula for Vertical Stretch or Compression If f is a function and k is a constant, then the graph of y = k · f(x) is the graph of y = f(x) • Vertically stretched by a factor of k, if k > 1. • Vertically compressed by a factor of k, if 0 < k < 1. • Vertically stretched or compressed by a factor |k| and reflected across x-axis, if k < 0.

Combining Stretches and Shifts Example 3 The function y = f(x) has the graph shown: Graph the function g(x) = − ½ f(x + 3) − 1. Solution To combine several transformations, always work from inside the parentheses outward. The graphs corresponding to each step are shown. Step 1: shift 3 units to the left Step 2: vertically compress by ½ Step 3: reflect about the x-axis Step 4: shift down 1 unit Step 1: y = f(x+3) Step 2: y = ½ f(x+3) Step 3: y =- ½ f(x+3) Step 4: y = -½ f(x+3) - 1