Q. 15 - Precalculus Enhanced with Graphing Utilities

Question

For the equation $3x^2-4y=12$, find the intercepts and check for symmetry.

Step-by-Step Solution

Answer

The x-intercepts are $\left(2,0\right),\left(-2,0\right)$ and y-intercept is $\left(0,-3\right)$.

The graph is symmetric with respect to the y-axis.

Step 1. Given information.

Consider the given question,

$3x^2-4y=12$

Substitute $y=0$ in the given equation,

$$\begin{gathered} 3x^2-4\left(0\right)=12 \\ {x}^2=\frac{12}{3} \\ {x}^2=4 \\ x=\pm 2 \end{gathered}$$

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

Step 2. Find the y -intercepts.

Substitute $x=0$ in the given equation,

$$\begin{gathered} 3{\left(0\right)}^2-4y=12 \\ -4y=12 \\ y=-\frac{12}{4} \\ y=-3 \end{gathered}$$

Therefore, the y-intercept is$\left(0,-3\right)$.

Replace y with $-y$ in the given equation,

$$\begin{gathered} 3x^2-4\left(-y\right)=12 \\ 3x^2+4y=12 \end{gathered}$$

As this equation is not equivalent to the original equation, then the graph is not symmetric with respect to the x-axis.

Step 2. Find the symmetry.

Replace x with $-x$ in the given equation,

$$\begin{gathered} 3{\left(-x\right)}^2-4y=12 \\ 3x^2-4y=12 \end{gathered}$$

As this equation is equivalent to the original equation, then the graph is not symmetric with respect to the y-axis.

Replace x with $-x$ and y with $-y$ in the given equation,

$$\begin{gathered} 3{\left(-x\right)}^2-4\left(-y\right)=12 \\ 3x^2+4y=12 \end{gathered}$$

As this equation is not equivalent to the original equation, then the graph is not symmetric with respect to the origin.