Chapter 2.2, Problem 107E

To determine

To identify: The symmetry with respect to x -axis, y -axis, and the origin and x -intercept (s) of polynomial function $g\left(x\right)$ using graphing utility to graph the function.

Answer to Problem 107E

The function $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ is not symmetric with respect tox -axis, y -axis and the origin. The x -intercepts of $g\left(x\right)$ are $\left(-1,0\right)$ , $\left(3,0\right)$ , and $\left(4.5,0\right)$ .

Explanation of Solution

Given data:

The given polynomial function $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ .

Method/ concept used:

The graph of the function is plotted using graphing calculator. The x -intercept of the function $f\left(x\right)$ is the point $\left(a,0\right)$ , where $f\left(a\right)=0$ .

Graph:

The graph of the polynomial function $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ is shown in Figure-1.

  

Interpretation of the Graph:

Since the graph of $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ is not identical about x- axis (both sides of x -axis) that implies the function is not symmetric about x -axis.

[Note: if replacing y by -y function $y=f\left(x\right)$ does not change then the function is symmetrical about x − axis, e.g., $y^2=ax$ ]

Since the graph of $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ is notidentical about y -axis (both sides of y -axis) that implies the function is not symmetricwith respect to y -axis.

[Note: if replacing xby −x function $y=f\left(x\right)$ does not change then the function is symmetrical about y - axis, e.g., $y=a{x}^2$ .]

Since near the origin the graph of $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ is not identical about x- axis and y -axis simultaneously) that implies the function is not symmetricwith respect to origin.

[Note: A function is said to be symmetric about the origin if $\left(x,y\right)$ is on the graph then $\left(-x,-y\right)$ is also, e.g., $y^2=x^2$ .]

Thus, the function $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ is not symmetric with respect tox -axis and the origin but is symmetric with respect to y -axis.

The graph of $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ intersects x -axis at points $\left(-1,0\right)$

  $\left(3,0\right)$ , and $\left(4.5,0\right)$ .

Hence, $\left(-1,0\right)$ , $\left(3,0\right)$ , and $\left(4.5,0\right)$arex -interceptsof polynomial function $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ .

Conclusion:

The function $g\left(x\right)=\frac{1}{5}{\left(x+1\right)}^2\left(x-3\right)\left(2x-9\right)$ is not symmetric with respect tox -axis, y -axis and the origin. Thex -interceptsof $g\left(x\right)$ are $\left(-1,0\right)$ , $\left(3,0\right)$ , and $\left(4.5,0\right)$ .