<< Chapter < Page | Chapter >> Page > |
For the equation $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}\mathrm{cos}(Bx+C)+D,$ what constants affect the range of the function and how do they affect the range?
The absolute value of the constant $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ (amplitude) increases the total range and the constant $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ (vertical shift) shifts the graph vertically.
How does the range of a translated sine function relate to the equation $\text{\hspace{0.17em}}y=A\text{\hspace{0.17em}}\mathrm{sin}(Bx+C)+D?$
How can the unit circle be used to construct the graph of $\text{\hspace{0.17em}}f(t)=\mathrm{sin}\text{\hspace{0.17em}}t?$
At the point where the terminal side of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ intersects the unit circle, you can determine that the $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the y -coordinate of the point.
For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y -values and their corresponding x -values on one period for $\text{\hspace{0.17em}}x>0.\text{\hspace{0.17em}}$ Round answers to two decimal places if necessary.
$f(x)=2\mathrm{sin}\text{\hspace{0.17em}}x$
$f(x)=\frac{2}{3}\mathrm{cos}\text{\hspace{0.17em}}x$
amplitude: $\text{\hspace{0.17em}}\frac{2}{3};\text{\hspace{0.17em}}$ period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=\frac{2}{3}\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-\frac{2}{3}\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\pi ;\text{\hspace{0.17em}}$ for one period, the graph starts at 0 and ends at $\text{\hspace{0.17em}}2\pi $
$f(x)=-3\mathrm{sin}\text{\hspace{0.17em}}x$
$f(x)=4\mathrm{sin}\text{\hspace{0.17em}}x$
amplitude: 4; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum $\text{\hspace{0.17em}}y=4\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\frac{\pi}{2};\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-4\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\frac{3\pi}{2};\text{\hspace{0.17em}}$ one full period occurs from $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=2\pi $
$f(x)=2\mathrm{cos}\text{\hspace{0.17em}}x$
$f\left(x\right)=\mathrm{cos}\left(2x\right)$
amplitude: 1; period: $\text{\hspace{0.17em}}\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=1\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\pi ;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-1\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\frac{\pi}{2};\text{\hspace{0.17em}}$ one full period is graphed from $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=\pi $
$f(x)=2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{1}{2}x\right)$
$f(x)=4\text{\hspace{0.17em}}\mathrm{cos}(\pi x)$
amplitude: 4; period: 2; midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=4\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-4\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=1$
$f(x)=3\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{6}{5}x\right)$
$y=3\text{\hspace{0.17em}}\mathrm{sin}(8(x+4))+5$
amplitude: 3; period: $\text{\hspace{0.17em}}\frac{\pi}{4};\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=5;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=8\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0.12;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=2\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0.516;\text{\hspace{0.17em}}$ horizontal shift: $\text{\hspace{0.17em}}-4;\text{\hspace{0.17em}}$ vertical translation 5; one period occurs from $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=\frac{\pi}{4}$
$y=2\text{\hspace{0.17em}}\mathrm{sin}(3x-21)+4$
$y=5\text{\hspace{0.17em}}\mathrm{sin}(5x+20)-2$
amplitude: 5; period: $\text{\hspace{0.17em}}\frac{2\pi}{5};\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=\mathrm{-2};\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=3\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0.08;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=\mathrm{-7}\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\mathrm{0.71;}\text{\hspace{0.17em}}$ phase shift: $\text{\hspace{0.17em}}\mathrm{-4};\text{\hspace{0.17em}}$ vertical translation: $\text{\hspace{0.17em}}\mathrm{-2;}\text{\hspace{0.17em}}$ one full period can be graphed on $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=\frac{2\pi}{5}$
For the following exercises, graph one full period of each function, starting at $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ For each function, state the amplitude, period, and midline. State the maximum and minimum y -values and their corresponding x -values on one period for $\text{\hspace{0.17em}}x>0.\text{\hspace{0.17em}}$ State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.
$f\left(t\right)=2\mathrm{sin}\left(t-\frac{5\pi}{6}\right)$
$f(t)=-\mathrm{cos}\left(t+\frac{\pi}{3}\right)+1$
amplitude: 1 ; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=1;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=2\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=2.09;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=2\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}t=2.09;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=0\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}t=5.24;\text{\hspace{0.17em}}$ phase shift: $\text{\hspace{0.17em}}-\frac{\pi}{3};\text{\hspace{0.17em}}$ vertical translation: 1; one full period is from $\text{\hspace{0.17em}}t=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}t=2\pi $
$f\left(t\right)=4\mathrm{cos}\left(2\left(t+\frac{\pi}{4}\right)\right)-3$
$f\left(t\right)=-\mathrm{sin}\left(\frac{1}{2}t+\frac{5\pi}{3}\right)$
amplitude: 1; period: $\text{\hspace{0.17em}}4\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=1\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}t=11.52;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-1\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}t=5.24;\text{\hspace{0.17em}}$ phase shift: $\text{\hspace{0.17em}}-\frac{10\pi}{3};\text{\hspace{0.17em}}$ vertical shift: 0
$f\left(x\right)=4\mathrm{sin}\left(\frac{\pi}{2}\left(x-3\right)\right)+7$
Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in [link] .
amplitude: 2; midline: $\text{\hspace{0.17em}}y=-3;\text{\hspace{0.17em}}$ period: 4; equation: $\text{\hspace{0.17em}}f(x)=2\mathrm{sin}\left(\frac{\pi}{2}x\right)-3$
Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .
Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .
amplitude: 2; period: 5; midline: $\text{\hspace{0.17em}}y=3;\text{\hspace{0.17em}}$ equation: $\text{\hspace{0.17em}}f(x)=-2\mathrm{cos}\left(\frac{2\pi}{5}x\right)+3$
Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link] .
Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .
amplitude: 4; period: 2; midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ equation: $\text{\hspace{0.17em}}f(x)=-4\mathrm{cos}\left(\pi \left(x-\frac{\pi}{2}\right)\right)$
Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link] .
Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .
amplitude: 2; period: 2; midline $\text{\hspace{0.17em}}y=1;\text{\hspace{0.17em}}$ equation: $\text{\hspace{0.17em}}f\left(x\right)=2\mathrm{cos}\left(\pi x\right)+1$
Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link] .
For the following exercises, let $\text{\hspace{0.17em}}f(x)=\mathrm{sin}\text{\hspace{0.17em}}x.$
On $\text{\hspace{0.17em}}[0,2\pi ),$ solve $\text{\hspace{0.17em}}f\left(x\right)=0.$
On $\text{\hspace{0.17em}}[0,2\pi ),$ solve $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{2}.$
$\frac{\pi}{6},\frac{5\pi}{6}$
Evaluate $\text{\hspace{0.17em}}f\left(\frac{\pi}{2}\right).$
On $\text{\hspace{0.17em}}[0,2\pi ),f(x)=\frac{\sqrt{2}}{2}.\text{\hspace{0.17em}}$ Find all values of $\text{\hspace{0.17em}}x.$
$\frac{\pi}{4},\frac{3\pi}{4}$
On $\text{\hspace{0.17em}}[0,2\pi ),$ the maximum value(s) of the function occur(s) at what x -value(s)?
On $\text{\hspace{0.17em}}[0,2\pi ),$ the minimum value(s) of the function occur(s) at what x -value(s)?
$\frac{3\pi}{2}$
Show that $\text{\hspace{0.17em}}f(-x)=-f(x).\text{\hspace{0.17em}}$ This means that $\text{\hspace{0.17em}}f(x)=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is an odd function and possesses symmetry with respect to ________________.
For the following exercises, let $\text{\hspace{0.17em}}f(x)=\mathrm{cos}\text{\hspace{0.17em}}x.$
On $\text{\hspace{0.17em}}[0,2\pi ),$ solve the equation $\text{\hspace{0.17em}}f(x)=\mathrm{cos}\text{\hspace{0.17em}}x=0.$
$\frac{\pi}{2},\frac{3\pi}{2}$
On $\text{\hspace{0.17em}}[0,2\pi ),$ solve $\text{\hspace{0.17em}}f(x)=\frac{1}{2}.$
On $\text{\hspace{0.17em}}[0,2\pi ),$ find the x -intercepts of $\text{\hspace{0.17em}}f(x)=\mathrm{cos}\text{\hspace{0.17em}}x.$
$\frac{\pi}{2},\frac{3\pi}{2}$
On $\text{\hspace{0.17em}}[0,2\pi ),$ find the x -values at which the function has a maximum or minimum value.
On $\text{\hspace{0.17em}}[0,2\pi ),$ solve the equation $\text{\hspace{0.17em}}f(x)=\frac{\sqrt{3}}{2}.$
$\frac{\pi}{6},\frac{11\pi}{6}$
Graph $\text{\hspace{0.17em}}h(x)=x+\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[0,2\pi \right].\text{\hspace{0.17em}}$ Explain why the graph appears as it does.
Graph $\text{\hspace{0.17em}}h(x)=x+\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[-100,100\right].\text{\hspace{0.17em}}$ Did the graph appear as predicted in the previous exercise?
The graph appears linear. The linear functions dominate the shape of the graph for large values of $\text{\hspace{0.17em}}x.$
Graph $\text{\hspace{0.17em}}f(x)=x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[0,2\pi \right]\text{\hspace{0.17em}}$ and verbalize how the graph varies from the graph of $\text{\hspace{0.17em}}f(x)=\mathrm{sin}\text{\hspace{0.17em}}x.$
Graph $\text{\hspace{0.17em}}f(x)=x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on the window $\text{\hspace{0.17em}}\left[\mathrm{-10},10\right]\text{\hspace{0.17em}}$ and explain what the graph shows.
The graph is symmetric with respect to the y -axis and there is no amplitude because the function is not periodic.
Graph $\text{\hspace{0.17em}}f(x)=\frac{\mathrm{sin}\text{\hspace{0.17em}}x}{x}\text{\hspace{0.17em}}$ on the window $\text{\hspace{0.17em}}\left[\mathrm{-5}\pi ,5\pi \right]\text{\hspace{0.17em}}$ and explain what the graph shows.
A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function $\text{\hspace{0.17em}}h\left(t\right)\text{\hspace{0.17em}}$ gives a person’s height in meters above the ground t minutes after the wheel begins to turn.
Notification Switch
Would you like to follow the 'Precalculus' conversation and receive update notifications?