# 10.2 Graphs of `y` = `a` sin `bx` and `y` = `a` cos `bx`

**Period of a Function • Graphs of $y\hspace{0.17em}=\hspace{0.17em}a\text{}\mathrm{sin}\text{}bx$ and $y\hspace{0.17em}=\hspace{0.17em}a\text{}\mathrm{cos}\text{}bx$ • Important Values for Sketching**

In graphing the function $y\hspace{0.17em}=\hspace{0.17em}\mathrm{sin}\text{}x\hspace{0.17em},\hspace{0.17em}$ we see that the values of `y` repeat every $2\pi $ units of `x`. This is because $\mathrm{sin}\text{}x\text{}\hspace{0.17em}=\hspace{0.17em}\mathrm{sin}(x\text{}\hspace{0.17em}+\hspace{0.17em}2\pi )\hspace{0.17em}=\hspace{0.17em}\mathrm{sin}(x\text{}\hspace{0.17em}+\hspace{0.17em}4\pi )\hspace{0.17em},\hspace{0.17em}$ and so forth. For any function `F`, we say that it has a *period P* if $F(x)\hspace{0.17em}=\hspace{0.17em}F(x\hspace{0.17em}+\hspace{0.17em}P)\hspace{0.17em}.\hspace{0.17em}$ For functions that are periodic, such as the sine and cosine, *the* **period** *is the x-distance between a point and the next corresponding point for which the value of y repeats.*

Let us now plot the curve $y\hspace{0.17em}=\hspace{0.17em}\mathrm{sin}\text{}2x\hspace{0.17em}.\hspace{0.17em}$ This means that we choose a value of `x`, multiply this value by 2, and find the sine of the result. This leads to the following table of values for this function: ...

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