Chapter 1.9, Problem 20E

To determine

To graph: The quadratic equation $y=\frac{x}{x^2+25}$ in a graphing window.

Explanation of Solution

Given information:

The quadratic equation $y=\frac{x}{x^2+25}$ .

Graph:

The graph of the quadratic equation $y=\frac{x}{x^2+25}$ can be sketched in the cartesian plane,

Consider the quadratic equation, $y=\frac{x}{x^2+25}$ .

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 numerator and denominator by $x$ in RHS:

  $\begin{array}{l}y=\frac{x}{x^2+25}\\ y=\frac{1}{x+\frac{25}{x}}\end{array}$

Logic:

  $a+\frac{k}{a}$ .

Minimum value:

  $\begin{array}{l}a=\frac{k}{a}\\ a^2=k\\ a=\sqrt{k}\end{array}$ .

Max $-\infty$ when $a$ is $\infty$ ,

  $\begin{array}{c}x+\frac{25}{x}\\ x=\frac{25}{x}\\ x^2=25\\ x=\sqrt{25}\end{array}$

The value of $x$ is $5$ .

Now substitute the value of $x=0$ in equation $y=\frac{x}{x^2+25}$ .

  $\begin{array}{l}y=\frac{x}{x^2+25}\\ y=\frac{0}{0^2+25}\\ y=0\end{array}$

Now put the value of $x=1$ in the equation, $y=\frac{x}{x^2+25}$ .

  $\begin{array}{l}y=\frac{x}{x^2+25}\\ y=\frac{1}{1^2+25}\\ y=\frac{1}{26}\end{array}$

Substitute the value $x=2$ in the equation $y=\frac{x}{x^2+25}$ :

  $\begin{array}{l}y=\frac{x}{x^2+25}\\ y=\frac{2}{2^2+25}\\ y=\frac{2}{4+25}\\ y=\frac{2}{29}\end{array}$

Substitute the value $x=3$ in the equation $y=\frac{x}{x^2+25}$ :

  $\begin{array}{l}y=\frac{x}{x^2+25}\\ y=\frac{3}{3^2+25}\\ y=\frac{3}{9+25}\\ y=\frac{3}{34}\end{array}$

Substitute the value $x=-1$ in the equation $y=\frac{x}{x^2+25}$ :

  $\begin{array}{l}y=\frac{x}{x^2+25}\\ y=\frac{-1}{-1^2+25}\\ y=\frac{-1}{24}\end{array}$

Substitute the value $x=-2$ in the equation $y=\frac{x}{x^2+25}$ :

  $\begin{array}{l}y=\frac{x}{x^2+25}\\ y=\frac{-2}{-2^2+25}\\ y=\frac{-2}{4+25}\\ y=\frac{-2}{29}\end{array}$

Now the value of $y$ is:

  $\begin{array}{l}y=\frac{-2}{4+25}\\ y=\frac{-2}{29}\\ y=-\frac{2}{29}\end{array}$

Substitute the value $x=-3$ in the equation $y=\frac{x}{x^2+25}$ :

  $\begin{array}{l}y=\frac{x}{x^2+25}\\ y=\frac{-3}{-3^2+25}\\ y=\frac{-3}{9+25}\\ y=\frac{-3}{34}\end{array}$

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 $y=\frac{x}{x^2+25}$ 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 $\frac{x}{x^2+25}$ .

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

  $\begin{array}{c}X\min =-0.6\\ X\max =0.6\\ Y\min =-0.6\\ Y\max =0.6\end{array}$

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 $y=\frac{x}{x^2+25}$ represents a wave graph.

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, $y=\frac{x}{x^2+25}$ there is no $x-$ intercepts.

Therefore, the equation $y=\frac{x}{x^2+25}$ is symmetric about the $y-$ axis Initially the graph of the function $y=\frac{x}{x^2+25}$ increases when increase the value of $x$ ,the graph is a parabola shaped graph.