Section 3.1 : Graphing
9. Determine the \(x\)-intercepts and the \(y\)-intercepts for the following equation.
\[y = {\left( {x + 3} \right)^2} - 8\]Show All Steps Hide All Steps
Start SolutionRecall that in order to find the \(y\)-intercept all we need to do is plug \(x = 0\) into the equation and solve for \(y\). Doing that for this equation gives,
\[\begin{align*}y & = {\left( {0 + 3} \right)^2} - 8\\ y & = 1\end{align*}\]The \(y\)-intercept for this equation is then the point : \(\left( {0,1} \right)\) .
Show Step 2Finding the \(x\)-intercept is similar to the \(y\)-intercept. All we do is plug in \(y = 0\) and solve for \(x\). Doing that for this equation gives,
\[\begin{align*}0 & = {\left( {x + 3} \right)^2} - 8\\ {\left( {x + 3} \right)^2} & = 8\\ x + 3 & = \pm \sqrt 8 \\ x & = - 3 \pm \sqrt 8 \end{align*}\]The \(x\)-intercepts for this equation are then the two points : \(\left( { - 3 - \sqrt 8 ,0} \right)\) and \(\left( { - 3 + \sqrt 8 ,0} \right)\) .
Don’t worry about the “messy” answers here. This kind of intercept will show up occasionally so we need to be able to deal with them when they do.