Write a formula for the function graphed in
[link] .
two possibilities:
$\text{\hspace{0.17em}}y=4\mathrm{sin}\left(\frac{\pi}{5}x-\frac{\pi}{5}\right)+4\text{\hspace{0.17em}}$ or
$\text{\hspace{0.17em}}y=-4\mathrm{sin}\left(\frac{\pi}{5}x+\frac{4\pi}{5}\right)+4$
Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.
we will let
$\text{\hspace{0.17em}}C=0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}D=0\text{\hspace{0.17em}}$ and work with a simplified form of the equations in the following examples.
Given the function
$\text{\hspace{0.17em}}y=A\mathrm{sin}\left(Bx\right),\text{\hspace{0.17em}}$ sketch its graph.
Identify the amplitude,
$\text{\hspace{0.17em}}\left|A\right|.$
Identify the period,
$\text{\hspace{0.17em}}P=\frac{2\pi}{\left|B\right|}.$
Start at the origin, with the function increasing to the right if
$\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is positive or decreasing if
$\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is negative.
At
$\text{\hspace{0.17em}}x=\frac{\pi}{2\left|B\right|}\text{\hspace{0.17em}}$ there is a local maximum for
$\text{\hspace{0.17em}}A>0\text{\hspace{0.17em}}$ or a minimum for
$\text{\hspace{0.17em}}A<0,\text{\hspace{0.17em}}$ with
$\text{\hspace{0.17em}}y=A.$
The curve returns to the
x -axis at
$\text{\hspace{0.17em}}x=\frac{\pi}{\left|B\right|}.$
There is a local minimum for
$\text{\hspace{0.17em}}A>0\text{\hspace{0.17em}}$ (maximum for
$\text{\hspace{0.17em}}A<0$ ) at
$\text{\hspace{0.17em}}x=\frac{3\pi}{2\left|B\right|}\text{\hspace{0.17em}}$ with
$\text{\hspace{0.17em}}y=\u2013A.$
The curve returns again to the
x -axis at
$\text{\hspace{0.17em}}x=\frac{\pi}{2\left|B\right|}.$
Graphing a function and identifying the amplitude and period
Sketch a graph of
$\text{\hspace{0.17em}}f\left(x\right)=-2\mathrm{sin}\left(\frac{\pi x}{2}\right).$
Let’s begin by comparing the equation to the form
$\text{\hspace{0.17em}}y=A\mathrm{sin}(Bx).$
Step 1. We can see from the equation that
$\text{\hspace{0.17em}}A=-2,$ so the amplitude is 2.
$\left|A\right|=2$
Step 2. The equation shows that
$\text{\hspace{0.17em}}B=\frac{\pi}{2},\text{\hspace{0.17em}}$ so the period is
Step 3. Because
$\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is negative, the graph descends as we move to the right of the origin.
Step 4–7. The
x -intercepts are at the beginning of one period,
$\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}$ the horizontal midpoints are at
$\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and at the end of one period at
$\text{\hspace{0.17em}}x=4.$
The quarter points include the minimum at
$\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and the maximum at
$\text{\hspace{0.17em}}x=3.\text{\hspace{0.17em}}$ A local minimum will occur 2 units below the midline, at
$\text{\hspace{0.17em}}x=1,\text{\hspace{0.17em}}$ and a local maximum will occur at 2 units above the midline, at
$\text{\hspace{0.17em}}x=3.\text{\hspace{0.17em}}$[link] shows the graph of the function.
Sketch a graph of
$\text{\hspace{0.17em}}g\left(x\right)=-0.8\mathrm{cos}\left(2x\right).\text{\hspace{0.17em}}$ Determine the midline, amplitude, period, and phase shift.
Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.
Express the function in the general form
$\text{\hspace{0.17em}}y=A\mathrm{sin}(Bx-C)+D\text{or}y=A\mathrm{cos}(Bx-C)+D.$
Identify the amplitude,
$\text{\hspace{0.17em}}\left|A\right|.$
Identify the period,
$\text{\hspace{0.17em}}P=\frac{2\pi}{\left|B\right|}.$
Identify the phase shift,
$\text{\hspace{0.17em}}\frac{C}{B}.$
Draw the graph of
$\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{sin}\left(Bx\right)\text{\hspace{0.17em}}$ shifted to the right or left by
$\text{\hspace{0.17em}}\frac{C}{B}\text{\hspace{0.17em}}$ and up or down by
$\text{\hspace{0.17em}}D.$
Graphing a transformed sinusoid
Sketch a graph of
$\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{sin}\left(\frac{\pi}{4}x-\frac{\pi}{4}\right).$
Step 1. The function is already written in general form:
$\text{\hspace{0.17em}}f(x)=3\mathrm{sin}\left(\frac{\pi}{4}x-\frac{\pi}{4}\right).$ This graph will have the shape of a
sine function , starting at the midline and increasing to the right.
Step 2.$\text{\hspace{0.17em}}\left|A\right|=\left|3\right|=3.\text{\hspace{0.17em}}$ The amplitude is 3.
Step 3. Since
$\text{\hspace{0.17em}}\left|B\right|=\left|\frac{\pi}{4}\right|=\frac{\pi}{4},\text{\hspace{0.17em}}$ we determine the period as follows.
Draw a graph of
$\text{\hspace{0.17em}}g(x)=-2\mathrm{cos}\left(\frac{\pi}{3}x+\frac{\pi}{6}\right).\text{\hspace{0.17em}}$ Determine the midline, amplitude, period, and phase shift.
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection