Q. 7 - Precalculus Enhanced with Graphing Utilities

Question

Sketch a graph of each of the following functions. Label at least three points on each graph. 

$\left(a\right) y=x^2, \left(b\right)\hspace{0.17em}y=x^3, \left(c\right) y=e^x, \left(d\right)\hspace{0.17em}y=\ln x, \left(e\right) y=\sin x, \left(f\right)\hspace{0.17em}y=\tan x$

Step-by-Step Solution

Answer

The graph of the equations are:

Part (a): 

Part (b) :

Part (c):

Part (d):

Part (e):

Part (f):

Part (a) Step 1: Given Information

The equation given is $y=x^2$. We need to sketch the graph and label three points on it.

Part (a) Step 2: Sketching and labelling the graph

Substituting the points $x=-1,0,1$ on the graph we get,

when $x=-1$,

style="width:30%" $$\begin{gathered} y=x^2 \\ \Rightarrow y={(-1)}^2 \\ \Rightarrow y=1 \end{gathered}$$

when $x=0$,

$$\begin{gathered} y=x^2 \\ \Rightarrow y={\left(0\right)}^2 \\ \Rightarrow y=0 \end{gathered}$$

when $x=1$,

$$\begin{gathered} y=x^2 \\ \Rightarrow y={\left(1\right)}^2 \\ \Rightarrow y=1 \end{gathered}$$

The points on the graph are $(-1,1), (0,0), (1,1)$.

The graph of the equation is:

Part (b) Step 1: Given Information

The equation given is $y=x^3$. We need to sketch the graph and label three points on it.

Part (b) Step 2: Sketching and labelling the graph

Substituting the points $x=-1, 0, 1$on the graph we get,

when $x=-1$,

style="width:30%" $$\begin{gathered} y=x^3 \\ \Rightarrow y={(-1)}^3 \\ \Rightarrow y=-1 \end{gathered}$$

when $x=0$,

style="width:30%" $$\begin{gathered} y=x^3 \\ \Rightarrow y={\left(0\right)}^3 \\ \Rightarrow y=0 \end{gathered}$$

when $x=1$,

$$\begin{gathered} y=x^3 \\ \Rightarrow y={\left(1\right)}^3 \\ \Rightarrow y=1 \end{gathered}$$

The points on the graph are $(-1,-1), (0,0), (1,1)$.

The graph of the equation is:

Part (c) Step 1: Given Information

The equation given is $y=e^x$. We need to sketch the graph and label three points on it.

Part (c) Step 2: Sketching and labelling the graph

Substituting the points $x=-1, 0, 1$on the graph we get,

when $x=-1$

$$\begin{gathered} y=e^x \\ \Rightarrow y=e^{-1} \\ \Rightarrow y=\frac{1}{e} \end{gathered}$$

when $x=0$

$$\begin{gathered} y=e^x \\ \Rightarrow y=e^0 \\ \Rightarrow y=1 \end{gathered}$$

when $x=1$,

$$\begin{gathered} y=e^x \\ \Rightarrow y=e^1 \\ \Rightarrow y=e \end{gathered}$$

The points on the graph are $\left(-1,\frac{1}{e}\right), (0,1), (1,e)$.

The graph of the equation is:

Part (d) Step 1: Given Information

The equation given is $y=\ln x$. We need to sketch the graph and label three points on it.

Part (d) Step 2: Sketching and labelling the graph

The equation can also be written as:

$$\begin{gathered} y=\ln x \\ \Rightarrow x=e^y \end{gathered}$$

Substituting the points $y=-1, 0, 1$on the graph we get,

when $y=-1$,

$$\begin{gathered} x=e^y \\ \Rightarrow x=e^{-1} \\ \Rightarrow x=\frac{1}{e} \end{gathered}$$

when style="width:30%" $y=0$,

$$\begin{gathered} x=e^y \\ \Rightarrow x=e^0 \\ \Rightarrow x=1 \end{gathered}$$

when $y=1$,

$$\begin{gathered} x=e^y \\ \Rightarrow x=e^1 \\ \Rightarrow x=e \end{gathered}$$

The points on the graph are $\left(\frac{1}{e},-1\right), (1,0), (e,1)$.

The graph of the equation is:

Part (e) Step 1: Given Information

The equation given is $y=\sin x$. We need to sketch the graph and label three points on it. 

Part (e) Step 2: Sketching and labelling the graph

Substituting the values of $x=-\frac{\mathrm{\pi }}{2}, 0, \frac{\mathrm{\pi }}{2}$ we get,

when $x=-\frac{\mathrm{\pi }}{2}$

$$\begin{gathered} y=\sin x \\ \Rightarrow y=\sin \left(-\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow y=-1 \end{gathered}$$

when $x=0$,

$$\begin{gathered} y=\sin x \\ \Rightarrow y=\sin \left(0\right) \\ \Rightarrow y=0 \end{gathered}$$

when $x=\frac{\mathrm{\pi }}{2}$

$$\begin{gathered} y=\sin x \\ \Rightarrow y=\sin \left(\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow y=1 \end{gathered}$$

The points on the graph are $\left(-\frac{\mathrm{\pi }}{2},-1\right), (0,0), \left(\frac{\mathrm{\pi }}{2},1\right)$.

The graph of the function is:

Part (f) Step 1: Given Information

The equation given is $y=\tan x$. We need to sketch the graph and label three points on it. 

Part (f) Step 2: Sketching and labelling the graph

Substituting the values of $x=-\frac{\mathrm{\pi }}{4}, 0, \frac{\mathrm{\pi }}{4}$on the graph we get,

when $x=-\frac{\mathrm{\pi }}{4}$,

$$\begin{gathered} y=\tan x \\ \Rightarrow y=\tan \left(-\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow y=-1 \end{gathered}$$

when $x=0$,

$$\begin{gathered} y=\tan x \\ \Rightarrow y=\tan \left(0\right) \\ \Rightarrow y=0 \end{gathered}$$

when $x=\frac{\mathrm{\pi }}{4}$,

$$\begin{gathered} y=\tan x \\ \Rightarrow y=\tan \left(\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow y=1 \end{gathered}$$

The points on the graph are $\left(-\frac{\mathrm{\pi }}{4},-1\right), (0,0), \left(\frac{\mathrm{\pi }}{4},1\right)$.

The graph of the equation is: