CPM Homework Help : PC3 Problem 5-74

Using the graph of $y=f(x)$ below, sketch a graph of each of the following transformations.

Continuous piecewise, 4 linear pieces, labeled, f of x, starting at lower left, passing through the point (negative 4, comma negative 2), turning horizontal at the point (negative 1, comma 1), turning down at the point (1, comma 1), turning horizontal at the point (5, comma negative 1), continuing to the right.

$y=−2f(x)$
Steps (a):
1. Reflect vertically (across the $x$ -axis).
2. Stretch vertically by a factor of $2$ .

$y=f(−x)+1$
Steps (b):
1. Reflect horizontally (across the $y$ -axis).
2. Shift up $1$ unit.

$y=\frac{1}{f(x)}$
Step 1(c):
Sketch the vertical asymptotes. Added to the graph, 2 vertical dashed lines at, x = negative 2, & at, x = 3.
Step 2(c):
Since $\frac{1}{1}=1$ and $\frac{1}{-1}=-1$ , mark the points that "stay in place" on the graph. The following Points highlighted: (negative 3, comma negative 1), (negative 1, comma 1), (0, comma 1), (1, comma 1), & (5, comma negative 1).
Step 3(c):
Where the original function is positive, the reciprocal function will also be positive.

Where the original function is negative, the reciprocal function will also be negative.