Q. 19 - Precalculus Enhanced with Graphing Utilities

Question

Sketch the graph of each function. Be sure to label three points on the graph. 

$f\left(x\right)=x^3$

Step-by-Step Solution

Answer

The graph is

Step 1. Given information.

The given function is

$f\left(x\right)=x^3$.

Step 2. Finding the symmetry.

Now substituting $x=-x$ in the given function then,

$$\begin{gathered} f(-x)={\left(-x\right)}^3 \\ =-x^3 \\ =-f\left(x\right) \end{gathered}$$

Therefore, the function is an odd function and symmetric with respect to the origin.

Step 3. Finding the intercepts.

On substituting $x=0$, the $y$-intercept will be

$$\begin{gathered} f\left(0\right)={\left(0\right)}^3 \\ =0 \end{gathered}$$

Then the $y$-intercept is $\left(0,0\right)$.

On substituting $f\left(x\right)=0$, the $x$-intercept will be

$$\begin{gathered} f\left(x\right)=0 \\ {x}^3=0 \\ x=0 \end{gathered}$$

Then the $x$-intercept is $\left(0,0\right)$.

Step 4. Finding the ordered pairs.

Using the point plotting method we get,

$x$	$$\begin{gathered} y=f\left(x\right) \\ =x^3 \end{gathered}$$	$\left(x,y\right)$
$-1$	$$\begin{gathered} y={\left(-1\right)}^3 \\ =-1 \end{gathered}$$	$\left(-1,-1\right)$
$0$	$$\begin{gathered} y={\left(0\right)}^3 \\ =0 \end{gathered}$$	$\left(0,0\right)$
$1$	$$\begin{gathered} y={\left(1\right)}^3 \\ =1 \end{gathered}$$	$\left(1,1\right)$

Step 5. Plotting the points on a graph.

The graph is