8.1 Graphs of the sine and cosine functions  (Page 7/13)

Key equations

Key concepts

• Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of $2\pi .$
• The function $\sin x$ is odd, so its graph is symmetric about the origin. The function $\cos x$ is even, so its graph is symmetric about the y -axis.
• The graph of a sinusoidal function has the same general shape as a sine or cosine function.
• In the general formula for a sinusoidal function, the period is $P=\frac{2\pi }{\left|B\right|}.$ See [link] .
• In the general formula for a sinusoidal function, $\left|A\right|$ represents amplitude. If $\left|A\right|>1,$ the function is stretched, whereas if $\left|A\right|<1,$ the function is compressed. See [link] .
• The value $\frac{C}{B}$ in the general formula for a sinusoidal function indicates the phase shift. See [link] .
• The value $D$ in the general formula for a sinusoidal function indicates the vertical shift from the midline. See [link] .
• Combinations of variations of sinusoidal functions can be detected from an equation. See [link] .
• The equation for a sinusoidal function can be determined from a graph. See [link] and [link] .
• A function can be graphed by identifying its amplitude and period. See [link] and [link] .
• A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See [link] .
• Sinusoidal functions can be used to solve real-world problems. See [link] , [link] , and [link] .

Section exercises

Verbal