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Exercises Trigonometry

Example 9 from Algebra and Trigonometry, 8.1 Graphs of the Sine and Cosine Functions

Graphing a Transformed Sinusoid

Sketch a graph of $f (x) = 3 \sin (\frac{π}{4} x - \frac{π}{4}) .$

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. The function is already written in general form: $f (x) = 3 \sin (\frac{π}{4} x - \frac{π}{4}) .$ This graph will have the shape of a sine function, starting at the midline and increasing to the right.
- Step 2. $| A | = | 3 | = 3.$ The amplitude is 3.
- Step 3. Since $| B | = | \frac{π}{4} | = \frac{π}{4},$ we determine the period as follows.
  $P = \frac{2 π}{| B |} = \frac{2 π}{\frac{π}{4}} = 2 π \cdot \frac{4}{π} = 8$
  
  The period is 8.
- Step 4. Since $C = \frac{π}{4},$ the phase shift is
  $\frac{C}{B} = \frac{\frac{π}{4}}{\frac{π}{4}} = 1.$
  
  The phase shift is 1 unit.
- Step 5. Figure 20 shows the graph of the function.
  
  Figure 20 A horizontally compressed, vertically stretched, and horizontally shifted sinusoid

Your turn

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

Draw a graph of $g (x) = - 2 \cos (\frac{π}{3} x + \frac{π}{6}) .$ Determine the midline, amplitude, period, and phase shift.

How did it go?

PreviousGraphing a Function and Identifying the Amplitude and Period Next Identifying the Properties of a Sinusoidal Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 8.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.