Q. 17 - Precalculus Enhanced with Graphing Utilities

Question

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$y=-x^2+4.$

Step-by-Step Solution

Answer

The intercepts are :

x-intercepts a$\left(-2,0\right)$ and $\left(2,0\right)$ ,

y-intercept is $\left(0,4\right)$.

Step 1: Given information:

Consider the equation,

$y=-x^2+4$.

Step 2: To find the x-intercept:

Let $y=0$ in the equation,

Then, $-x^2+4=0$

$\left(-1\right)\times \left(-x^2+4\right)=\left(-1\right)\times 0$ [Multiply by $\left(-1\right)$ on both sides.]

$x^2-4=0$

$\left(x+2\right)\left(x-2\right)=0$ [Factors]

$x+2=0$  or  $x-2=0$ [Zero Product Property]

$x=-2$  or  $x=2$

The equation has two solutions, $-2 and 2.$

$x-intercepts are \left(-2,0\right) and \left(2,0\right).$

Step 3: To find the y-intercept:

Let $x=0$ in the equation,

Then, $y=0+4$

$y=4$

$y-intercept is \left(0,4\right).$

Step 4: To draw the graph:

We can draw the graph of the equation $y=-x^2+4$ by plotting the above intercepts and the points in the table below,

$x$	$y=-x^2+4$	$\left(x,y\right)$
$-3$	$-5$	$\left(-3,-5\right)$
$-1$	$3$	$\left(-1,3\right)$
$1$	$3$	$\left(1,3\right)$
$3$	$-5$