Q. 21 - Precalculus Enhanced with Graphing Utilities

Question

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

$9x^2+4y=36$.

Step-by-Step Solution

Answer

The intercepts are :

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

y-intercept is $\left(0,9\right)$.

Step 1: Given information:

Consider the equation,

$9x^2+4y=36$

Step 2: To find the x-intercept:

Let $y=0$ in the equation,

Then, $9x^2+4\times 0=36$

$9x^2=36$

$\frac{9x^2}{9}=\frac{36}{9}$ [Divide by 9 on both sides.]

$x^2=4$

$x^2-4=4-4$  [Add $\left(-4\right)$ 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 2: To find the y-intercept:

Let $x=0$ in the equation,

Then, $9\times 0+4y=36$

$4y=36$

$\frac{4y}{4}=\frac{36}{4}$

$y=9$

$y-intercept is (0,9).$

Step 4: To draw the graph:

We can draw the graph of the equation $9x^2+4y=36$ by plotting the above intercepts and the points in the table below,

$x$	$y=\frac{36-9x^2}{4}$	$\left(x,y\right)$
$-3$	$\frac{-45}{4}$	$\left(-3,\frac{-45}{4}\right)$
$-1$	$\frac{27}{4}$	$\left(-1,\frac{27}{4}\right)$
$1$	$\frac{27}{4}$	$\left(1,\frac{27}{4}\right)$
$3$	$\frac{-45}{4}$	$\left(3,\frac{-45}{4}\right)$