Basic Technical Mathematics, 11th Edition

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

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

In graphing the function $y\hspace{0.17em}=\hspace{0.17em}\sin x\hspace{0.17em},\hspace{0.17em}$ we see that the values of y repeat every $2\pi$ units of x. This is because $\sin x\hspace{0.17em}=\hspace{0.17em}\sin (x\hspace{0.17em}+\hspace{0.17em}2\pi )\hspace{0.17em}=\hspace{0.17em}\sin (x\hspace{0.17em}+\hspace{0.17em}4\pi )\hspace{0.17em},\hspace{0.17em}$ 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 $y\hspace{0.17em}=\hspace{0.17em}\sin 2x\hspace{0.17em}.\hspace{0.17em}$ 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: ...