Chapter 5, Problem 1PRE

For Exercises 1—16, identify which functions shown here (f, g, h, and so on) have the given characteristics.

$\begin{array}{l}f\left(x\right)=-3\sec \left(2x+\pi \right)g\left(x\right)=-3\cos \left(\frac{1}{2}x-\frac{\pi }{4}\right)h\left(x\right)=3\sin \left(-\frac{1}{2}x-\frac{\pi }{4}\right)\\ k\left(x\right)=\sin \left(\frac{\pi }{2}x\right)+3m\left(x\right)=2\csc \left(2x-\frac{\pi }{4}\right)-3n\left(x\right)=3\tan \left(x-\frac{\pi }{2}\right)\\ p\left(x\right)=-2\cot \left(\frac{1}{2}x+\pi \right)t\left(x\right)=-3+2\cos x\end{array}$

1. Has an amplitude of 3

To determine

Toidentify : the functions that have the given characteristics

Answer to Problem 1PRE

The functions that have amplitude of 3 are: $g\left(x\right)=-3\cos \left(\frac{1}{2}x-\frac{\pi }{4}\right)$ , & $h\left(x\right)=3\sin \left(-\frac{1}{2}x-\frac{\pi }{4}\right)$

Explanation of Solution

Given information:Listed functions in below

$f\left(x\right)=\sin \left(\frac{\pi }{2}x\right)+3$	$g\left(x\right)=-3\cos \left(\frac{1}{2}x-\frac{\pi }{4}\right)$	$h\left(x\right)=3\sin \left(-\frac{1}{2}x-\frac{\pi }{4}\right)$
$k\left(x\right)=-3\sec \left(2x+\pi \right)$	$m\left(x\right)=2\csc \left(2x-\frac{\pi }{2}\right)-3$	$n\left(x\right)=3\tan \left(x-\frac{\pi }{2}\right)$
$p\left(x\right)=-2\cot \left(\frac{1}{2}x+\pi \right)$	$t\left(x\right)=-3+2\cos x$

Concept Involved:

Properties of the General Sine and Cosine Functions:

Consider the graphs of $y=A\sin \left(Bx-C\right)+D$ and $y=A\cos \left(Bx-C\right)+D$ with $B>0$ .

• The amplitude is |A|.
• The period is $\frac{2\pi }{B}$ .
• The phase shift is $\frac{C}{B}$ .
• The vertical shift is D.
• One full cycle is given on the interval $0\le Bx-C\le 2\pi$ .
• The domain is $-\infty <x<\infty$ .
• The range is $-\left|A\right|+D\le y\le \left|A\right|+D$ .

Properties of the tangent and cotangent functions $y=A\tan \left(Bx-C\right)+D$ and $y=A\cot \left(Bx-C\right)+D$ with $B>0$ (If $B<0$ , we can rewrite the equations using the odd function property of the tangent and cotangent functions.)

• The value $\left|A\right|$ is the vertical scaling factor.
• If $A<0$, the graph is reflected across the x-axis
• The period is $\frac{\pi }{B}$
• The phase shift is $\frac{C}{B}$
• The vertical shift is D

Calculation:

Function	Period	Phase Shift	Vertical shift	Amplitude
$f\left(x\right)=\sin \left(\frac{\pi }{2}x\right)+3$	$\frac{2\pi }{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\frac{4\pi }{\pi }=4$	0	3	1
$g\left(x\right)=-3\cos \left(\frac{1}{2}x-\frac{\pi }{4}\right)$	$\frac{2\pi }{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\frac{4\pi }{1}=4\pi$	$\frac{\pi }{4}÷\frac{1}{2}=\frac{2\pi }{4}=\frac{\pi }{2}$	0	3
$h\left(x\right)=3\sin \left(-\frac{1}{2}x-\frac{\pi }{4}\right)$	$\frac{2\pi }{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\frac{4\pi }{1}=4\pi$	$-\frac{\pi }{4}÷\frac{1}{2}=-\frac{2\pi }{4}=-\frac{\pi }{2}$	0	3
$k\left(x\right)=-3\sec \left(2x+\pi \right)$	$\frac{2\pi }{2}=\pi$	$-\frac{\pi }{2}$	0	None
$m\left(x\right)=2\csc \left(2x-\frac{\pi }{2}\right)-3$	$\frac{2\pi }{2}=\pi$	$\frac{\pi }{2}÷\frac{2}{1}=\frac{\pi }{2}\cdot \frac{1}{2}=\frac{\pi }{4}$	-3	None
$n\left(x\right)=3\tan \left(x-\frac{\pi }{2}\right)$	$\frac{\pi }{1}=\pi$	$\frac{\pi }{2}÷1=\frac{\pi }{2}$	0	None
$p\left(x\right)=-2\cot \left(\frac{1}{2}x+\pi \right)$	$\frac{\pi }{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\frac{\pi }{1}\cdot \frac{2}{1}=2\pi$	$-\pi ÷\frac{1}{2}=-\pi \cdot \frac{2}{1}=-2\pi$	0	None
$t\left(x\right)=-3+2\cos x$	$\frac{2\pi }{1}=2\pi$	0	-3	2