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Section 6.3 Graphing Parabolas

Subsection 1. Find the coordinates of points on a parabola

To find the \(x\)-coordinate of a point on a parabola, we usually need to solve a quadratic equation.

Subsubsection Examples

Example 6.39.

Find the \(y\)-coordinate of the point on the graph of \(~y=2x^2-3x+5~\) with \(x\)-coordinate \(-3\text{.}\)

Solution

Substitute \(x=\alert{-3}\) into the equation, and evaluate.

\begin{equation*} y=2(\alert{-3})^2-3(\alert{-3})+5 = 18+9+5 = 32 \end{equation*}

The \(y\)-coordinate is 32, and the point is \((-3,32)\text{.}\)

Example 6.40.

Find the \(x\)-coordinates of all points on the graph of \(~y=20-3x^2~\) with \(y\)-coordinate \(-28\text{.}\)

Solution

Substitute \(y=\alert{-28}\) into the equation, and solve.

\begin{align*} \alert{-28} \amp = 20-3x^2 \amp \amp \blert{\text{Subtract 20 from both sides.}}\\ -48 \amp =-3x^2 \amp \amp \blert{\text{Divide both sides by}~-3.}\\ 16 \amp = x^2 \amp \amp \blert{\text{Extract roots.}}\\ \pm 4 \amp = x \end{align*}

The points are \((4, -28)\) and \((-4,-28)\text{.}\)

Subsubsection Exercises

Find the \(y\)-coordinate of the point on the graph of \(~y=-x^2+6x+2~\) with \(x\)-coordinate \(-2\text{.}\)

Answer
\(-14\)

The \(x\)-coordinate of the vertex of \(~y=2x^2-6x+1~\) is \(\dfrac{3}{2}\text{.}\) Find the \(y\)-coordinate of the vertex.

Answer
\(\dfrac{-7}{2}\)

Find the \(x\)-coordinates of all points on the graph of \(~y=x^2-2x+5~\) with \(y\)-coordinate 8.

Answer
\(-1,~3\)

Find the \(x\)-intercepts of the graph of \(~y=\dfrac{1}{4}x^2-5x+24~\text{.}\)

Answer
\((8,0),~(12,0)\)

Subsection 2. Find the average of two numbers

The average of two numbers lies half-way between them on a number line. To find their average, we take one-half of their sum. That is, the average of \(p\) and \(q\) is

\begin{equation*} \dfrac{1}{2}(p+q)~~~~\text{or}~~~~\dfrac{p+q}{2} \end{equation*}

Subsubsection Example

Example 6.45.

The average of 4 and 9 is

\begin{equation*} \dfrac{1}{2}(4+9) = \dfrac{1}{2}(13) = \dfrac{13}{2},~~\text{or}~~6\dfrac{1}{2} \end{equation*}
Example 6.46.

The average of \(-8\) and \(4\) is

\begin{equation*} \dfrac{1}{2}(-8+4) = \dfrac{1}{2}(-4) = -2 \end{equation*}
Example 6.47.

The average of \(\dfrac{5}{2}\) and \(\dfrac{-3}{4}\) is

\begin{equation*} \dfrac{1}{2}\left(\dfrac{5}{2}-\dfrac{3}{4}\right) = \dfrac{1}{2}\left(\dfrac{10}{4}-\dfrac{3}{4}\right) = \dfrac{1}{2}\left(\dfrac{10}{4}\right) = \dfrac{7}{8} \end{equation*}

Subsubsection Exercises

Find the average of \(-12\) and \(-7\text{.}\)

Answer
\(\dfrac{-19}{2}\)

Find the average of \(-4\) and \(\dfrac{1}{2}\text{.}\)

Answer
\(\dfrac{-7}{4}\)

Find the average of \(\dfrac{3}{2}\) and \(\dfrac{9}{2}\text{.}\)

Answer
\(3\)

Find the average of \(\dfrac{9}{4}\) and \(\dfrac{-3}{4}\text{.}\)

Answer
\(\dfrac{3}{4}\)