Chapter 1.9, Problem 19E

To determine

To graph: The quadratic equation $y=x^4-4x^3$ in a graphing window.

Explanation of Solution

Given information:

The quadratic equation $y=x^4-4x^3$ .

Graph:

The graph of the quadratic equation $y=x^4-4x^3$ can be sketched in the cartesian plane,

Consider the quadratic equation, $y=x^4-4x^3$ .

Now put the values of $x$ in the equation to find the values of $y$ or to find the real roots of $y$ with the help of $x$ .

Rewrite the equation as take common:

  $\begin{array}{l}y=x^4-4x^3\\ y=x^3\left(x-4\right)\end{array}$

Now substitute the value $y=0$ in the equation $y=x^4-4x^3$ .

  $\begin{array}{l}y=x^3\left(x-4\right)\\ 0=x^3\left(x-4\right)\end{array}$

Either $x^3=0$ or $x-4=0$ .

Hence $x=0$ and $x=4$ .

Now substitute the value of $x=0$ in equation $y=x^4-4x^3$ .

  $\begin{array}{l}y=x^4-4x^3\\ y=0^4-4\cdot 0^3\\ y=0\end{array}$

Now put the value of $x=1$ in the equation, $y=x^4-4x^3$ .

  $\begin{array}{l}y=x^4-4x^3\\ y=1^{{}^4}-4\cdot 1^3\\ y=1-4\\ y=-3\end{array}$

Substitute the value $x=2$ in the equation $y=x^4-4x^3$ :

  $\begin{array}{l}y=x^4-4x^3\\ y=2^4-4\cdot 2^3\\ y=16-4\cdot 8\\ y=16-32\end{array}$

Now the value of $y$ is:

  $\begin{array}{l}y=16-4\cdot 8\\ y=16-32\\ y=-16\end{array}$

Substitute the value $x=3$ in the equation $y=x^4-4x^3$ :

  $\begin{array}{l}y=x^4-4x^3\\ y=3^4-4\cdot 3^3\\ y=81-4\cdot 27\\ y=81-108\end{array}$

Now the value of $y$ is:

  $\begin{array}{l}y=81-4\cdot 27\\ y=81-108\\ y=-27\end{array}$

Substitute the value $x=-1$ in the equation $y=x^4-4x^3$ :

  $\begin{array}{l}y=x^4-4x^3\\ y=-1^4-4\cdot -1^3\\ y=1+4\\ y=5\end{array}$

Substitute the value $x=-2$ in the equation $y=x^4-4x^3$ :

  $\begin{array}{l}y=x^4-4x^3\\ y=-2^4-4\cdot -2^3\\ y=16+4\cdot 8\\ y=16+32\end{array}$

Now the value of $y$ is:

  $\begin{array}{l}y=16+4\cdot 8\\ y=16+32\\ y=48\end{array}$

Substitute the value $x=-3$ in the equation $y=x^4-4x^3$ :

  $\begin{array}{l}y=x^4-4x^3\\ y=-3^4-4\cdot -3^3\\ y=81-4\cdot -27\\ y=81+108\end{array}$

Now the value of $y$ is:

  $\begin{array}{l}y=81-4\cdot -27\\ y=81+108\\ y=189\end{array}$

Observed that when the value of $x$ increases then the value of $y$ decreases slightly.

When the value of $x$ decreases then the value of $y$ increases slightly.

Steps to plot the graph of the equation $y=x^4-4x^3$ with the help of graphing utility are as follows:

Step 1: Press $\boxed{\text{MODE}}$ key.

Step 2: Use the down arrow key to reach $\boxed{\text{FUNC}}$ option.

Step 3: Press $\boxed{\text{ENTER}}$ key.

Step 4: Press $\boxed{\text{Y=}}$ key.

Step 5: Enter the function $y=x^4-4x^3$ .

Step 6: Press $\boxed{\text{WINDOW}}$ key. Change the settings to

  $\begin{array}{c}X\min =-10\\ X\max =10\\ Y\min =-10\\ Y\max =10\end{array}$

For better view of graph.

Step 8: Press $\boxed{\text{GRAPH}}$ key.

The result obtained on the screen is provided below,

  

Interpretation:

The equation of the function $y=x^4-4x^3$ represents a parabola.

The parabola opens downwards.

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

At $x$ -axis $y$ is always zero.

Recall that the graphical approach to solve the equation simultaneously.

Therefore, in the equation, $y=x^4-4x^3$ there is no $x-$ intercepts.

Therefore, the equation $y=x^4-4x^3$ is symmetric about the $y-$ axis Initially the graph of the function $y=x^4-4x^3$ increases when increase the value of $x$ ,the graph is a parabola shaped graph.