Question

An equation is given.

(a)Use a graphing calculator to graph the equation in the given viewing rectangle. (b) Find the x- and y-intercepts from the graph.

(c) Verify your answers to part (b) algebraically (from the equation).

$y=x^3-x^2;\left[-2,2\right]\text{by}\left[-1,1\right]$

Short answer

(a) The graph of the equation $y=x^3-x^2$is provided below.

(b) The x-intercept is $1$ and y-intercept is $0$.

(c) The x-intercepts are $0\text{and 1}$ and y-intercept is $0$.

Step-by-step solution

Step 1.  Apply the concept of coordinates.

The set of all points $\left(x,y\right)$ in the Cartesian plane or coordinate plane that satisfy the equation is known as the graph of the equation in x and y coordinates.

Step 2. Follow the steps of TI-83

Consider the provided equation $y=x^3-x^2$and scale of axes $\left[-2,2\right]\text{by}\left[-1,1\right]$.

Open the TI-83 calculator and follow the steps below.

Step 1: Press $\text{Y}=$key.

Step 2: Enter the function ${\text{Y}}_1=x\hat{}3-x\hat{}2$.

Step 3: Press the WINDOW key.

Step 4: Make the following changes.

$\begin{array}{c}X\min =-2\\ X\max =2\\ Y\min =-1\\ Y\max =1\end{array}$

Step 5: Press the TRACE key.

Step 3. Graph the equation

The result obtained on screen is the graph of the equation $y=x^3-x^2$.

Step 1.  Graph the equation

Open the TI-83 calculator and follow the steps below.

Step 1: Press $\text{Y}=$key.

Step 2: Enter the function ${\text{Y}}_1=x\hat{}3-x\hat{}2$.

Step 3: Press the WINDOW key.

Step 4: Make the following changes.

$\begin{array}{c}X\min =-100\\ X\max =150\\ Y\min =-2000\\ Y\max =2000\end{array}$

Step 5: Press the TRACE key.

Step 2. Find the x-intercept

Once, the graph is obtained on screen, follow the steps below to compute the x-intercept.

Step 1: Press 2nd key and then TRACE key.

Step 2: Select the second option “zero”.

Step 3: To enter the left bound, move the curve to the left of the x-intercept and press ENTER key.

Step 4: To enter the right bound, move the curve to the left of the x-intercept and press ENTER key.

Step 5: For a guess value, press ENTER key.

The result obtained on screen is provided below.

Therefore, x-intercept is$1$.

Step 3.  Find the y-intercept

Once, the graph is obtained on screen, follow the steps below to compute the y-intercept.

Step 1: Press 2nd key and then TRACE key.

Step 2: Select the first option “value”.

Step 3: Enter the x value as 0.

Step 4: Press ENTER key.

The result obtained on screen is provided below.

Therefore, y-intercept is $0$.

Step 1. Apply the concept of intercepts

The x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

The y-intercepts are the points on y-axis where the graph of the equation intersects the y-axis.

Step 2. Compute the x-intercept

Consider the equation $y=x^3-x^2$.

To find the x-intercept, substitute $y=0$in the equation $y=x^3-x^2$.

$\begin{array}{c}0=x^3-x^2\\ 0=x^2\left(x-1\right)\\ x=0\text{and}x-1=0\\ x=0\text{and}x=1\end{array}$

Therefore, x-intercepts are $0\text{and 1}$.

Step 3. Compute the y-intercept

Consider the equation $y=x^3-x^2$.

To find the y-intercept, substitute $x=0$in the equation $y=x^3-x^2$.

$\begin{array}{c}y=0^3-0^2\\ y=0\end{array}$

Therefore, y-intercept is $0$.

Hence, the x-intercepts are $0\text{and 1}$ and y-intercept is $0$.