Alg-tool Graphing Parabolas

Section 3.3 Graphing Parabolas

Subsection 1. 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 3.38.

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 3.39.

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

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

Example 3.40.

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

Checkpoint 3.41.

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

Answer

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

Checkpoint 3.42.

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

Answer

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

Checkpoint 3.43.

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

Answer

\(3\)

Checkpoint 3.44.

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

Answer

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

Subsection 2. Solve quadratic equations

Subsubsection Examples

Example 3.45.

Solve \(~3x^2=48\)

Solution

We use extraction of roots. We first divide by 3 to isolate the squared expression.

\begin{align*} x^2 \amp = 16 \amp \amp \blert{\text{Take square roots.}}\\ x \amp = \pm 4 \end{align*}

The solutions are \(x=4\) and \(x=-4\text{.}\)

Example 3.46.

Solve \(~3x^2=12x\)

Solution

We solve by factoring. First, we get zero on one side of the equation.

\begin{align*} 3x^2-12x \amp = 0 \amp \amp \blert{\text{Factor the left side.}}\\ 3x(x-4) \amp = 0 \amp \amp \blert{\text{Set each factor equal to zero.}}\\ 3x=0,~~~~x-4 \amp = 0 \end{align*}

The solutions are \(x=0\) and \(x=4\text{.}\)

Example 3.47.

Solve \(~3x^2-10x-8=0\)

Solution

We solve by factoring. We factor the left side.

\begin{align*} (x-4)(3x+2) \amp = 0 \amp \amp \blert{\text{Set each factor equal to zero.}}\\ x-4=0,~~~~3x+2 \amp = 0 \amp \amp \blert{\text{Solve each equation.}}\\ x=4,~~~~x\amp = \dfrac{-2}{3} \end{align*}

The solutions are \(x=0\) and \(x=\dfrac{-2}{3}\text{.}\)

Subsubsection Exercises

Checkpoint 3.48.

Solve \(~5x^2-30=0\)

Answer

\(x=\pm \sqrt{6}\)

Checkpoint 3.49.

Solve \(~\dfrac{1}{3}(x-2)^2=8\)

Answer

\(x=2 \pm \sqrt{24}\)

Checkpoint 3.50.

Solve \(~x^2-5x=300\)

Answer

\(x=20,~x=-15\)

Checkpoint 3.51.

Solve \(~4x^2+13x-12=0\)

Answer

\(x=\dfrac{3}{4},~x=-4\)

Subsection 3. 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 3.52.

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 3.53.

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 \((-4,-28)\text{.}\)