Q 6.102 - Intermediate Algebra

Question

For the function $f\left(x\right)=6x^2 +13x-15,$ find

(a) the zeros of the function

(b) any x-intercepts of the graph of the function

(c) any y-intercepts of the graph of the function.

Step-by-Step Solution

Answer

Part a. The zeros of the function are $\frac{5}{6}, -3.$

Part b. The x-intercepts of the graph of the functions are $(\frac{5}{6},0) and (-3,0).$

Part c. The y-intercept of the graph of the function is $(0,-15).$

Part a Step 1. Given Information

The given function is $f\left(x\right)=6x^2+13x-15$.

We have to find the zeros of the function.

Part a Step 2. Finding the zeros of the function

To find the zeros of the function, substitute $0$ for $f\left(x\right)$.

So, $$\begin{gathered} f\left(x\right)=6x^2+13x-15 \\ 0=6x^2+13x-15 \end{gathered}$$

Now, by factoring the trinomial we get

$0=(6x-5)(x+3)$

Therefore, the zeros of the function are

 $$\begin{gathered} 6x-5=0 \\ x=\frac{5}{6} \end{gathered}$$ and $$\begin{gathered} x+3=0 \\ x=-3 \end{gathered}$$

Part b Step 1. Given Information

The given function is $f\left(x\right)=6x^2+13x-15.$

We have to find any x-intercepts of the graph of the function.

Part b Step 2. Finding x-intercepts of the graph of the function

The x-intercept occurs when $y=0$

As we depict that the zeros of the function are $x=\frac{5}{6},-3$

Therefore, $$\begin{gathered} f\left(\frac{5}{6}\right)=0 \\ f(-3)=0 \end{gathered}$$

Thus, the points $(\frac{5}{6},0) and (-3,0)$ lie on the graph and these points are the x-intercepts of the function.

Part c Step 1. Given Information

The given function is $f\left(x\right)=6x^2+13x-15.$

We have to find the y-intercepts of the graph of the function. 

Part c Step 2. Finding y-intercepts of the graph of the function

The y-intercept occurs when $x=0$

So, substitute $0$ for x in f(x)

$$\begin{gathered} f\left(0\right)=6{\left(0\right)}^2+13\left(0\right)-15 \\ f\left(0\right)=0+0-15 \\ f\left(0\right)=-15 \end{gathered}$$

Thus, the point $(0, -15)$ lies on the graph and this point is the y-intercept of the function.