Q. 439 - Intermediate Algebra

Question

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.

$f\left(x\right)=64x^2-49$

Step-by-Step Solution

Answer

(a) The zeroes of the function are $x=-\frac{7}{8},\frac{7}{8}$

(b) The $x$-intercepts of the graph of the function are $\left(-\frac{7}{8},0\right),\left(\frac{7}{8},0\right)$

(c) The $y$-intercepts of the graph of the function are $(0,-49)$

Step 1. Given

The function is $f\left(x\right)=64x^2-49$

Step 2. (a) Find the zeroes of the function

For zeroes of the function, substitute $f\left(x\right)=0$.

$$\begin{gathered} f\left(x\right)=64x^2-49 \\ 0=64x^2-49 \\ (8x+7)(8x-7)=0 \end{gathered}$$

So,

$x=\frac{7}{8},-\frac{7}{8}$

Step 3. (b) Find the x -intercepts of the graph of the function.

Put $f\left(x\right)=0$ to find $x$-intercept.

$f\left(x\right)$ is $0$ at $x=\frac{7}{8},-\frac{7}{8}$

The $x$-intercept of the graph of the function are $\left(\frac{7}{8},0\right)$ and $\left(-\frac{7}{8},0\right)$

Step 4. (c) Find the y -intercepts of the graph of the function.

Put $x=0$ to find the $y$-intercept.

$$\begin{gathered} f\left(x\right)=64x^2-49 \\ f\left(0\right)=64{\left(0\right)}^2-49 \\ f\left(0\right)=-49 \end{gathered}$$

The $y$-intercept of the graph of the function is $(0,-49)$