Q. 319 - Intermediate Algebra

Question

In the following exercises, for each function, find:

a. the zeros of the function

b. the $x$- intercepts of the function

c. the $y$- intercepts of the function

$f\left(x\right)=6x^2-7x-5$

Step-by-Step Solution

Answer

a. The zeros are $\frac{5}{3}, -\frac{1}{2}$.

b. The $x$- intercepts are $(\frac{5}{3}, 0)$ and $(-\frac{1}{2}, 0)$.

c. The y-intercept is $(0, -5)$.

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.

$$\begin{gathered} f\left(x\right)=6x^2-7x-5 \\ 0=6x^2-7x-5 \\ 6x^2-7x-5=0 \\ 6x^2-10x+3x-5=0 \\ 2x(3x-5)+1(3x-5)=0 \\ (3x-5)(2x+1)=0 \end{gathered}$$

Using the Zero Product Property,

$3x-5=0$ or

$2x+1=0$.

When $3x-5=0$,

$x=\frac{5}{3}$

When $2x+1=0$,

$x=-\frac{1}{2}$.

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,

$f\left(\frac{5}{3}\right)=0$ and

$f(-\frac{1}{2})=0$.

Thus, the points $(\frac{5}{3}, 0)$ and $(-\frac{1}{2}, 0)$ 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 $f\left(0\right)$.

$$\begin{gathered} f\left(x\right)=6x^2-7x-5 \\ f\left(0\right)=6{\left(0\right)}^2-7\left(0\right)-5 \\ f\left(0\right)=-5 \end{gathered}$$

Since $f\left(0\right)=-5$, the point $(0, -5)$ lies on the graph and is the $y$- intercept of the function.