7.2 The General Sinusoidal Function
Algebra Refresher
- Write a formula for [latex]g(x){.}[/latex]
- Graph [latex]f(x)[/latex] and [latex]g(x)[/latex] on the same axes.
- [latex]\displaystyle f(x)=x^2,~g(x)=f(x-2)[/latex]
- [latex]\displaystyle f(x)=\sqrt{x},~g(x)=f(x+4)[/latex]
- [latex]\displaystyle f(x)=\dfrac{1}{x},~g(x)=f(x+1)[/latex]
- [latex]\displaystyle f(x)=\dfrac{1}{x^2},~g(x)=f(x-1)[/latex]
- [latex]\displaystyle f(x)=\lvert {x} \rvert,~g(x)=f(x-3)[/latex]
- [latex]\displaystyle f(x)=2^x,~g(x)=f(x+3)[/latex]
Algebra Refresher Answers
- [latex]\displaystyle g(x)=(x-2)^2[/latex]
- [latex]\displaystyle g(x)=\sqrt{x+4}[/latex]
- [latex]\displaystyle g(x)=\dfrac{1}{x+1}[/latex]
- [latex]\displaystyle g(x)=\dfrac{1}{(x-1)^2}[/latex]
- [latex]\displaystyle g(x)=\\vert {x-3} \rvert[/latex]
- [latex]\displaystyle g(x)=2^{x+3}[/latex]
- Graph trigonometric functions using a table of values
- Find a formula for a transformation of a trigonometric function
- Solve trigonometric equations graphically
- Model periodic phenomena with trigonometric functions
- Fit a circular function to data
Horizontal Shifts
In the previous section, we considered transformations of sinusoidal graphs, including vertical shifts, which change the midline of the graph; vertical stretches and compressions, which change its amplitude; and horizontal stretches and compressions, which occur when we change the period of the graph.
In this section, we consider one more transformation: shifting the graph horizontally. The figure below shows four different transformations of the graph of [latex]y=\sin x{.}[/latex]
Graph [latex]f(x)=\sin x[/latex] and [latex]g(x)= \sin \left(x - \dfrac{\pi}{4}\right)[/latex] in the ZTrig window. How is the graph of [latex]g[/latex] different from the graph of [latex]f{?}[/latex]
Solution
The graphs are shown below. The graph of [latex]g(x)= \sin (x - \dfrac{\pi}{4})[/latex] has the same amplitude, midline, and period as the graph of [latex]f(x)=\sin x{,}[/latex] but the graph of [latex]g[/latex] is shifted to the \right by [latex]\dfrac{\pi}{4}[/latex] units, compared to the graph of [latex]f{.}[/latex]
We can see why this shift occurs by studying a table of values for the two functions.
| [latex]x[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{4}[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\dfrac{3\pi}{4}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{5\pi}{4}[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]\dfrac{7\pi}{4}[/latex] |
[latex]2\pi[/latex] |
| [latex]f(x)[/latex] |
[latex]0[/latex] |
[latex]\dfrac{sqrt{2}}{2}[/latex] |
[latex]1[/latex] |
[latex]\dfrac{sqrt{2}}{2}[/latex] |
[latex]0[/latex] |
[latex]\dfrac{-sqrt{2}}{2}[/latex] |
[latex]-1[/latex] |
[latex]\dfrac{-sqrt{2}}{2}[/latex] |
[latex]0[/latex] |
| [latex]g(x)[/latex] |
[latex]\dfrac{-sqrt{2}}{2}[/latex] |
[latex]0[/latex] |
[latex]\dfrac{sqrt{2}}{2}[/latex] |
[latex]1[/latex] |
[latex]\dfrac{sqrt{2}}{2}[/latex] |
[latex]0[/latex] |
[latex]\dfrac{-sqrt{2}}{2}[/latex] |