Chapter 1.9, Problem 17E

To determine

To graph: The quadratic equation $y=0.01x^3-x^2+5$ in a graphing window.

Explanation of Solution

Given information:

The quadratic equation $y=0.01x^3-x^2+5$ .

Graph:

The graph of the quadratic equation $y=0.01x^3-x^2+5$ can be sketched in the cartesian plane,

Consider the quadratic equation, $y=0.01x^3-x^2+5$ .

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 take common:

  $\begin{array}{l}y=0.01x^3-x^2+5\\ y=0.01\left(x^3-100x^2+500\right)\end{array}$

Now substitute the value of $y$ in equation $y=0.01\left(x^3-100x^2+500\right)$ .

  $\begin{array}{l}y=0.01\left(x^3-100x^2+500\right)\\ 0=0.01\left(x^3-100x^2+500\right)\\ 0=\left(x^3-100x^2+500\right)\end{array}$

Eliminate the quadratic term by substituting:

  $\begin{array}{l}y=x+\frac{100}{3}\\ x=y-\end{array}$

  $\begin{array}{l}y=x+\frac{100}{3}\\ x=y-\frac{100}{3}\end{array}$

Now put the value of $x$ in the equation, $\left(x^3-100x^2+500\right)=0$ .

  $\begin{array}{l}\left(x^3-100x^2+500\right)=0\\ {\left(y-\frac{100}{3}\right)}^3-100{\left(y-\frac{100}{3}\right)}^2+500=0\end{array}$

The value of $x$ -intercepts are:

  $\begin{array}{l}x\approx 2.21174271\\ x\approx ~2.2617928\end{array}$

Substitute the value $x=0$ in the equation $y=0.01x^3-x^2+5$ :

  $\begin{array}{l}y=0.01x^3-x^2+5\\ y=0.01{\left(0\right)}^3-0^2+5\\ y=5\end{array}$

Substitute the value $x=1$ in the equation $y=0.01x^3-x^2+5$ :

  $\begin{array}{l}y=0.01x^3-x^2+5\\ y=0.01{\left(1\right)}^3-1^2+5\\ y=0.01-1+5\\ y=4.01\end{array}$

Substitute the value $x=2$ in the equation $y=0.01x^3-x^2+5$ :

  $\begin{array}{l}y=0.01x^3-x^2+5\\ y=0.01{\left(2\right)}^3-2^2+5\\ y=0.01\cdot 8-4+5\\ y=1.08\end{array}$

Substitute the value $x=-1$ in the equation $y=0.01x^3-x^2+5$ :

  $\begin{array}{l}y=0.01x^3-x^2+5\\ y=0.01{\left(-1\right)}^3-{\left(-1\right)}^2+5\\ y=-0.01+1+5\\ y=5.99\end{array}$

Substitute the value $x=-2$ in the equation $y=0.01x^3-x^2+5$ :

  $\begin{array}{l}y=0.01x^3-x^2+5\\ y=0.01{\left(-2\right)}^3-{\left(-2\right)}^2+5\\ y=0.01\cdot 8+4+5\\ y=9.08\end{array}$

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 $y=0.01x^3-x^2+5$ 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 $0.01x^3-x^2+5$ .

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 $y=0.01x^3-x^2+5$ 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, $y=0.01x^3-x^2+5$ there is no $x-$ intercepts.

Therefore, the equation $y=0.01x^3-x^2+5$ is symmetric about the $y-$ axis Initially the graph of the function $y=0.01x^3-x^2+5$ increases when increase the value of $x$ ,the graph is a parabola shaped graph.