Chapter 4.5, Problem 77E

To determine

To write: an equation for a function with the given characteristics.

Answer to Problem 77E

  $f\left(x\right)=\cos \left(2x+\pi \right)-\frac{3}{2}$ .

Explanation of Solution

Given information:

A cosine curve with a period of $\pi$ , an amplitude of 1, a left phase shift of $\pi$ , and vertical translation down $\frac{3}{2}$ units.

Concept used:

For the standard equation is $y=a\cos \left(bx-c\right)+d$ .

Amplitude is $\left|a\right|$ and period is $\frac{2\pi }{b}$ .

Vertical shift is $d$ unit and horizontal shift is $\frac{c}{b}$

The value $c$ represents the horizontal shift. If $c$ is positive then the graph will be shifted by $c$ units towards right compared to the graph of $\sin x$ and if it is negative then the shift will be towards left.

Calculation:

The graph of $g\left(x\right)=\cos x$ has a maximum value or the amplitude as 1 the minimum value is $-1$ and has a period of $2\pi$ .

Therefore, for a cosine curve with a period of $\pi$ , an amplitude of 1, a left phase shift of $\pi$ , and a vertical translation up $\frac{3}{2}$ unit, we have to multiply $\cos x$ by 1, $x$ by 2 and add $2\pi$ to $2x$ .

Hence, the function should be $f\left(x\right)=\cos \left(2x+\pi \right)-\frac{3}{2}$ .