1. VERTICAL STRETCHES AND COMPRESSIONS
6.3
VERTICAL STRETCHES AND COMPRESSIONS

2. 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

3. Example: Vertical Compression

4. 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)

5. 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.

6. 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