Q. 18 - Precalculus Enhanced with Graphing Utilities

Question

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

$y=-x^2+1$.

Step-by-Step Solution

Answer

The intercepts are :

x-intercepts are $\left(-1,0\right)$ and $\left(1,0\right)$ ,

y-intercept is $\left(0,1\right)$.

Step 1: Given information:

Consider the equation,

$y=-x^2+1$.

Step 2: To find the x-intercept:

Let $y=0$ in the equation,

Then, $-x^2+1=0$

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

$x^2-1=0$

$\left(x+1\right)\left(x-1\right)=0$ [Factors]

$x+1=0$  or  $x-1=0$ [Zero Product Property]

$x=-1$  or   $x=1$

The equation has two solutions, $-1 and 1.$

$x-intercepts are \left(-1,0\right) and \left(1,0\right).$

Step 3: To find the y-intercept:

Let $x=0$ in the equation,

Then, $y=0+1$

$y=1$

$y-intercept is \left(0,1\right)$.

Step 4: To draw the graph:

We can draw the graph of the equation $y=-x^2+1$ by plotting the above intercepts and the points in the table below,

$x$	$y=-x^2+1$	$\left(x,y\right)$
$-3$	$-8$	$\left(-3,-8\right)$
$-2$	$-3$	$\left(-2,-3\right)$
$2$	$-3$	$\left(2,-3\right)$
$3$	$-8$	$\left(3,-8\right)$