Suppose
\(f(x)=\dfrac{3x-7}{x^2-9}\text{,}\) and we want to find its x- and y-intercepts.
Letβs start with the x-intercepts. Remember, that is where the graph crosses the
\(x\)-axis, which is where
\(y=0\text{.}\) So, our strategy is to set
\(y=0\) and solve for
\(x\text{:}\)
\begin{align*}
0 \amp= \dfrac{3x-7}{x^2-9}\\
{\color{red}{(x^2-9)}}0 \amp= \dfrac{3x-7}{x^2-9}{\color{red}{(x^2-9)}}\\
0 \amp= 3x-7\\
0 {\color{red}{+7}} \amp= 3x-7{\color{red}{+7}} \\
7\amp=3x\\
\dfrac{7}{\color{red}{3}}\amp=\dfrac{3x}{\color{red}{3}}\\
\dfrac{7}{3}\amp=x
\end{align*}
Since we always write our intercepts as ordered pairs, we know that our x-intercept is
\(\left(\frac{7}{3},0\right)\text{.}\)
Now, letβs find our y-intercept. Remember, that is where the graph crosses the
\(y\)-axis, which is where
\(x=0\text{.}\) So, our strategy is to set
\(x=0\) and solve for
\(y\text{:}\)
\begin{equation*}
y= \dfrac{3(0)-7}{(0)^2-9} = \dfrac{-7}{-9}=\dfrac{7}{9}
\end{equation*}
Since we always write our intercepts as ordered pairs, we know that our y-intercept is
\(\left(0,\frac{7}{9}\right)\text{.}\)