Q. 317 - Intermediate Algebra

Question

In the following exercises, for each function, find:

a. the zeros of the function

b. the $x$- intercepts of the graph of the function.

c. the  $y$- intercept of the graph of the function.

$f\left(x\right)=9x^2-4$

Step-by-Step Solution

Answer

a. The zeros are $\frac{2}{3},-\frac{2}{3}$.

b. The $x$- intercepts are $(\frac{2}{3}, 0)$ and $(-\frac{2}{3}, 0)$.

c. The $y$- intercept is $(0, -4)$.

Part a. Finding zeros of the function

To find the zeros of the function, we need to find when the function value is $0$.

We substitute $0$ for $f\left(x\right)$ in the given equation.

$$\begin{gathered} f\left(x\right)=9x^2-4 \\ 0=9x^2-4 \\ 9x^2=4 \\ {x}^2=\frac{4}{9} \\ x=\frac{2}{3} \end{gathered}$$

or $x=-\frac{2}{3}$

These are the zeros of the function.

Part b. The x - intercepts of the graph of the function

An $x$- intercept occurs when $y=0$.

We know that,

$f\left(\frac{2}{3}\right)=0$ and

$f(-\frac{2}{3})=0$.

Thus, the points $(\frac{2}{3}, 0)$ and $(-\frac{2}{3}, 0)$ lie on the graph of the function.

Hence, these points are the $x$- intercepts of the function.

Part c. The y - intercepts of the graph of the function

A $y$- intercept of the function occurs when $x=0$.

To find it, we find $f\left(0\right)$.

$$\begin{gathered} f\left(x\right)=9x^2-4 \\ f\left(0\right)=9{\left(0\right)}^2-4 \\ f\left(0\right)=-4 \end{gathered}$$

Since $f\left(0\right)=-4$, the point $(0,-4)$ lies on the graph and is the $y$- intercept of the function.