Chapter 2, Problem 15CR

To determine

To find: The intercepts for the equation $3x^2-4y=12$ and check for the symmetry.

Answer to Problem 15CR

Solution:

$y\text{-intercept}$: $\left(0,-3\right)$.

$x\text{-intercept}$: $\left(2,0\right)$ and $\left(-2,0\right)$.

Symmetric about $y\text{-axis}$.

Explanation of Solution

Given:

$3x^2-4y=12$

Formula used:

$x\text{-intercept}$ is a point on the graph of the equation, where the value of $y$ is zero.

$y\text{-intercept}$ is a point on the graph of the equation, where the value of $x$ is zero.

Calculation:

It is given that the equation is $3x^2-4y=12$.

Rearranging the given equation, we get,

$-4y=-3x^2+12$

$y=\frac{3x^2-12}{4}$

$y=\frac{3}{4}{x}^2-3$

Equating the above equation to zero, we get,

$0=\frac{3}{4}{x}^2-3$

$12=3x^2$

$x^2=4$

$x=\pm 2$

Thus, the $x\text{-intercepts}$ are $\left(2,0\right)$ and $\left(-2,0\right)$. This implies that the graph is symmetric about $y\text{-axis}$.

Now substituting the value of slope and $x$ in the given equation, we get,

$3x^2-4y=12$

$0-4y=12$

$y=-3$

Thus, the $y\text{-intercept}$ is $\left(0,-3\right)$.

The $x$ and $y\text{-intercepts}$ are shown in the graph and the given equation is symmetric about $y\text{-axis}$.