Q. 9 - Precalculus Enhanced with Graphing Utilities

Question

f$\left(x\right)$ = sin x

(a) What is the y-intercept of the graph of f ?

(b) For what numbers x, $-\mathrm{\pi }\le \mathrm{x}\le \mathrm{\pi }$, is the graph of f

increasing?

(c) What is the absolute maximum of f ?

(d) For what numbers x, $0\le x\le 2\mathrm{\pi }$, does f$\left(x\right)$ = 0?

(e) For what numbers x,$-2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi }$ , does $f\left(x\right)$ = 1?

Where does $f\left(x\right)$ = -1?

(f) For what numbers x,$-2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi }$ , does $f\left(x\right)$ = $-\frac{1}{2}$

(g) What are the x-intercepts of f ?

Step-by-Step Solution

Answer

(a) The y-intercept is $0$.

(b) 

for increasing function $x \in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$

(c) The absolute maximum is $1.$

(d) $f\left(x\right) = 0 for x\in \{0,\mathrm{\pi },2\mathrm{\pi }\}$

(e) $$\begin{gathered} f\left(x\right)=1 for \\ x=-\frac{3\mathrm{\pi }}{2} and x=\frac{\mathrm{\pi }}{2} \\ f\left(x\right)=-1 for \\ x=-\frac{\mathrm{\pi }}{2} and x=\frac{3\mathrm{\pi }}{2} \end{gathered}$$

(f) $$\begin{gathered} f\left(x\right)=\frac{1}{2} for \\ x=-\frac{5\mathrm{\pi }}{6},-\frac{\mathrm{\pi }}{6},\frac{7\mathrm{\pi }}{6},\frac{11\mathrm{\pi }}{6} \end{gathered}$$

(g) The x intercept is $n\mathrm{\pi }, \mathrm{n}\in \mathrm{I}$

Part (a) Step 1. Given Information

$f\left(x\right) = \sin x$

Part (a) Step 2. Calculation

Since the graph of $f\left(x\right)=\sin x$ passes through (0,0) therefore the y-intercept of the graph is 0.

Part (b) Step 1. Calculation

The graph of $f\left(x\right)=\sin \left(x\right)$is increasing from $x=\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$ as the slope of the graph is increasing.

Part (c) Step 1. Calculation

The amplitude of the $f\left(x\right)=\sin x is \pm 1$ therefore the absolute value is 1.

Part (d) Step 1. Calculation

$$\begin{gathered} f\left(x\right) = \sin x \\ therefore f\left(x\right) = 0 \\ for x\in \{0,\mathrm{\pi },2\mathrm{\pi }\} \end{gathered}$$

Part (e) Step 1. Calculation

$$\begin{gathered} f\left(x\right)=1 \\ for \\ x=-\frac{3\mathrm{\pi }}{2} and x=\frac{\mathrm{\pi }}{2} \end{gathered}$$

Part (f) Step 1. Calculation

$$\begin{gathered} f\left(x\right)=\frac{1}{2} \\ for \\ x=-\frac{5\mathrm{\pi }}{6},-\frac{\mathrm{\pi }}{6},\frac{7\mathrm{\pi }}{6},\frac{11\mathrm{\pi }}{6} \end{gathered}$$

Part (g) Step 1. Calculation

The x-intercept is $n\mathrm{\pi }, \mathrm{n}\in \mathrm{I}$.