Example 7 from Algebra and Trigonometry, 8.2 Graphs of the Other Trigonometric Functions

Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant

Sketch a graph of $y = 2 \csc (\frac{π}{2} x) + 1.$ What are the domain and range of this function?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
- Step 1. Express the function given in the form $y = 2 \csc (\frac{π}{2} x) + 1.$
- Step 2. Identify the stretching/compressing factor, $| A | = 2.$
- Step 3. The period is $\frac{2 π}{| B |} = \frac{2 π}{\frac{π}{2}} = \frac{2 π}{1} \cdot \frac{2}{π} = 4.$
- Step 4. The phase shift is $\frac{0}{\frac{π}{2}} = 0.$
- Step 5. Draw the graph of $y = A \csc (B x)$ but shift it up $D = 1.$
- Step 6. Sketch the vertical asymptotes, which occur at $x = 0, x = 2, x = 4.$
The graph for this function is shown in Figure 11.

Figure 11 A transformed cosecant function

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Given the graph of $f (x) = 2 \cos (\frac{π}{2} x) + 1$ shown in Figure 12, sketch the graph of $g (x) = 2 \sec (\frac{π}{2} x) + 1$ on the same axes.

A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.

Figure 12

How did it go?

Solution

Your turn

Solution

Your turn