Chapter 1, Problem 5RE

To determine

The symmetry with respect to both axes and the origin for the equation, $y=-3x+7$. Also sketch the graph of the equation.

The equation, $y=-3x+7$, is not symmetric about the $x$ -axis, the $y$ -axis, and the origin.

Explanation:

Consider the equation,

$y=-3x+7$.

To check for symmetry about $y$ -axis, replace $x$ with $-x$ in the original equation, and simplify the equation.

If the equation after simplifying is equivalent to the original equation, then the given equation will be symmetric about $y$ -axis.

Replace $x$ with $-x$ in the original equation as,

$\begin{array}{c}y=-3\left(-x\right)+7\\ =3x+7\end{array}$

Since the new equation is not equivalent to the original equation, the given equation is not symmetric about $y$ -axis.

To check for symmetry about $x$ -axis, replace $y$ with $-y$ in the original equation, and simplify the equation.

If the equation after simplifying is equivalent to the original equation, then the given equation will be symmetric about $x$ -axis.

Replace $y$ with $-y$ in the original equation

$\begin{array}{c}-y=-3x+7\\ y=3x-7\end{array}$

Since the new equation is not equivalent to the original equation, the given equation is not symmetric about $x$ -axis.

To check for symmetry about the origin, replace $x$ with $-x$, and $y$ with$-y$ in the original equation, and simplify the equation.

If the equation after simplifying is equivalent to the original equation, then the given equation will be symmetric about the origin.

Replace $x$ with $-x$ and $y$ with$-y$ in the original equation as,

$\begin{array}{c}-y=-3\left(-x\right)+7\\ -y=3x+7\\ y=-3x-7\end{array}$

Since the new equation is not equivalent to the original equation, the given equation is not symmetric about the origin.

Plot the graph of the equation, $y=-3x+7$, by using the table below which consists of the different values of $y$ for the different values of $x$.

$\begin{array}{|cccccc|}\hline y=-3x+7& 16& 13& 7& 1& -5\\ x& -3& -2& 0& 2& 4\\ \hline\end{array}$

Now plot the points,