Q. 22 - Precalculus Enhanced with Graphing Utilities

Question

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

$4x^2+y=4$.

Step-by-Step Solution

Answer

The intercepts are :

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

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

Step 1: Given information:

Consider the equation,

$4x^2+y=4$.

Step 2: To find the x-intercept:

Let $y=0$ in the equation,

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

$4x^2=4$

$\frac{4x^2}{4}=\frac{4}{4}$ [Divide by 4 on both sides.]

$x^2=1$

$x^2-1=1-1$   [Add $-1$ 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 2: To find the y-intercept:

Let $x=0$ in the equation,

Then, $4\times 0+y=4$

$y=4$

$y-intercept is (0,4)$.

Step 4: To draw the graph:

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

$x$	$y=4-4x^2$	$\left(x,y\right)$
$-3$	$-32$	$\left(-3,-32\right)$
$-2$	$-12$	$\left(-2,-12\right)$
$2$	$-12$	$\left(2,12\right)$
$3$	$-32$	$\left(3,-32\right)$