The profit on your school fundraiser is represented by the quadratic expression \begin{align*}-5p^2 + 400p - 8000\end{align*}, where \begin{align*}p\end{align*} is your price point. What price point will result in a maximum profit and what is that profit?

### Vertex Intercept and Standard Form

So far, we have only used the **standard form** of a quadratic equation, \begin{align*}y=ax^2+bx+c\end{align*} to graph a parabola. From standard form, we can find the vertex and either factor or use the Quadratic Formula to find the \begin{align*}x-\end{align*}intercepts. The **intercept form** of a quadratic equation is \begin{align*}y=a(x-p)(x-q)\end{align*}, where \begin{align*}a\end{align*} is the same value as in standard form, and \begin{align*}p\end{align*} and \begin{align*}q\end{align*} are the \begin{align*}x-\end{align*}intercepts. This form looks very similar to a factored quadratic equation.

Let's change \begin{align*}y=2x^2+9x+10\end{align*} to intercept form and find the vertex. Then we will graph the parabola.

First, let’s change this equation into intercept form by factoring. \begin{align*}ac = 20\end{align*} and the factors of 20 that add up to 9 are 4 and 5. Expand the \begin{align*}x-\end{align*}term.

\begin{align*}y &=2x^2+9x+10\\ y &=2x^2+4x+5x+10\\ y &=2x(x+2)+5(x+2)\\ y &=(2x+5)(x+2)\end{align*}

Notice, this does not exactly look like the definition. The factors cannot have a number in front of \begin{align*}x\end{align*}. Pull out the 2 from the first factor to get \begin{align*}y=2\left(x+\frac{5}{2}\right)(x+2)\end{align*}. Now, let's find the vertex. Recall that all parabolas are symmetrical. This means that the axis of symmetry is *halfway* between the \begin{align*}x-\end{align*}intercepts or their average.

\begin{align*}\text{axis of symmetry} = \frac{p+q}{2}= \frac{-\frac{5}{2}-2}{2}= -\frac{9}{2} \div 2 = -\frac{9}{2}\cdot\frac{1}{2}=-\frac{9}{4}\end{align*}

This is also the \begin{align*}x-\end{align*}coordinate of the vertex. To find the \begin{align*}y-\end{align*}coordinate, plug the \begin{align*}x-\end{align*}value into either form of the quadratic equation. We will use Intercept form.

\begin{align*}y &=2\left(-\frac{9}{4}+\frac{5}{2}\right)\left(-\frac{9}{4}+2\right)\\ y &=2\cdot\frac{1}{4}\cdot-\frac{1}{4}\\ y &=-\frac{1}{8}\end{align*}

The vertex is \begin{align*}\left(-2\frac{1}{4}, -\frac{1}{8}\right)\end{align*}. Now, we will plot the \begin{align*}x-\end{align*}intercepts and the vertex to graph.

The last form is vertex form. **Vertex form** is written \begin{align*}y=a(x-h)^2+k\end{align*}, where \begin{align*}(h, k)\end{align*} is the vertex and \begin{align*}a\end{align*} is the same is in the other two forms. Notice that \begin{align*}h\end{align*} is negative in the equation, but positive when written in coordinates of the vertex.

Now, let's find the vertex of \begin{align*}y=\frac{1}{2}(x-1)^2+3\end{align*} and graph the parabola.

The vertex is going to be (1, 3). To graph this parabola, use the symmetric properties of the function. Pick a value on the left side of the vertex. If \begin{align*}x = -3\end{align*}, then \begin{align*}y=\frac{1}{2}(-3-1)^2+3=11.\end{align*} -3 is 4 units away from 1 (the \begin{align*}x-\end{align*}coordinate of the vertex). 4 units on the *other* side of 1 is 5. Therefore, the \begin{align*}y-\end{align*}coordinate will be 11. Plot (1, 3), (-3, 11), and (5, 11) to graph the parabola.

Finally, let's change \begin{align*}y=x^2-10x+16\end{align*} into vertex form.

To change an equation from standard form into vertex form, you must complete the square. The major difference is that you will not need to solve this equation.

\begin{align*}y &=x^2-10x+16\\ y-16+{\color{red}25} &=x^2-10x+{\color{red}25} \quad \text{Move 16 to the other side and add} \ \left(\frac{b}{2}\right)^2 \text{to both sides}.\\ y+9 &=(x-5)^2 \qquad \quad \ \ \text{Simplify left side and factor the right side}\\ y &=(x-5)^2-9 \qquad \text{Subtract 9 from both sides to get} \ y \ \text{by itself}.\end{align*}

To solve an equation in vertex form, set \begin{align*}y = 0\end{align*} and solve for \begin{align*}x\end{align*}.

\begin{align*}(x-5)^2-9 &=0\\ (x-5)^2 &=9\\ x-5 &=\pm 3\\ x &=5 \pm 3 \ or \ 8 \ and \ 2\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find the price point that will result in a maximum profit and to find that profit.

The vertex will give us the price point that will result in the maximum profit and that profit, so let’s change this equation into intercept form by factoring. First factor out -5.

\begin{align*}-5p^2 + 400p - 8000 = -5(p^2 - 80p + 1600)\\ -5(p - 40)(p - 40)\end{align*}

From this we can see that the \begin{align*}x\end{align*}-intercepts are 40 and 40. The average of 40 and 40 is 40 (of course). Plug 40 into the original equation:

\begin{align*}-5(40)^2 + 400(40) - 8000 = -8000 + 16000 - 8000 = 0\end{align*}

Therefore, the price point that results in a maximum profit is $40 and that price point results in a profit of $0. You're not making any money, so you better rethink your fundraising approach!

#### Example 2

Find the intercepts of \begin{align*}y=2(x-7)(x+2)\end{align*} and change it to standard form.

The intercepts are the opposite sign from the factors; (7, 0) and (-2, 0). To change the equation into standard form, FOIL the factors and distribute \begin{align*}a\end{align*}.

\begin{align*}y &=2(x-7)(x+2)\\ y &=2(x^2-5x-14)\\ y &=2x^2-10x-28\end{align*}

#### Example 3

Find the vertex of \begin{align*}y = -\frac{1}{2}(x+4)^2-5\end{align*} and change it to standard form.

The vertex is (-4, -5). To change the equation into standard form, FOIL \begin{align*}(x+4)^2\end{align*}, distribute \begin{align*}a\end{align*}, and then subtract 5.

\begin{align*}y &=-\frac{1}{2}(x+4)(x+4)-5\\ y &=-\frac{1}{2}(x^2+8x+16)-5\\ y &=-\frac{1}{2}x^2-4x-21\end{align*}

#### Example 4

Change \begin{align*}y=x^2+18x+45\end{align*} to intercept form and graph.

To change \begin{align*}y=x^2+18x+45\end{align*} into intercept form, factor the equation. The factors of 45 that add up to 18 are 15 and 3. Intercept form would be \begin{align*}y = (x+15)(x+3)\end{align*}. The intercepts are (-15, 0) and (-3, 0). The \begin{align*}x-\end{align*}coordinate of the vertex is halfway between -15 and -3, or -9. The \begin{align*}y-\end{align*}coordinate of the vertex is \begin{align*}y=(-9)^2+18(-9)+45=-36\end{align*}. Here is the graph:

#### Example 5

Change \begin{align*}y=x^2-6x-7\end{align*} to vertex form and graph.

To change \begin{align*}y=x^2-6x-7\end{align*} into vertex form, complete the square.

\begin{align*}y+7+9 &=x^2-6x+9\\ y+16 &=(x-3)^2 \\ y &=(x-3)^2-16\end{align*}

The vertex is (3, -16).

For vertex form, we could solve the equation by using square roots or we could factor the standard form. Either way, we will get that the \begin{align*}x-\end{align*}intercepts are (7, 0) and (-1, 0).

### Review

- Fill in the table below. Either describe how to find each entry or use a formula.

Find the vertex and \begin{align*}x\end{align*}-intercepts of each function below. Then, graph the function. If a function does not have any \begin{align*}x\end{align*}-intercepts, use the symmetry property of parabolas to find points on the graph.

- \begin{align*}y=(x-4)^2-9\end{align*}
- \begin{align*}y=(x+6)(x-8)\end{align*}
- \begin{align*}y=x^2+2x-8\end{align*}
- \begin{align*}y=-(x-5)(x+7)\end{align*}
- \begin{align*}y=2(x+1)^2-3\end{align*}
- \begin{align*}y=3(x-2)^2+4\end{align*}
- \begin{align*}y=\frac{1}{3}(x-9)(x+3)\end{align*}
- \begin{align*}y=-(x+2)^2+7\end{align*}
- \begin{align*}y=4x^2-13x-12\end{align*}

Change the following equations to intercept form.

- \begin{align*}y=x^2-3x+2\end{align*}
- \begin{align*}y=-x^2-10x+24\end{align*}
- \begin{align*}y=4x^2+18x+8\end{align*}

Change the following equations to vertex form.

- \begin{align*}y=x^2+12x-28\end{align*}
- \begin{align*}y=-x^2-10x+24\end{align*}
- \begin{align*}y=2x^2-8x+15\end{align*}

Change the following equations to standard form.

- \begin{align*}y=(x-3)^2+8\end{align*}
- \begin{align*}y=2\left(x-\frac{3}{2}\right)(x-4)\end{align*}
- \begin{align*}y=-\frac{1}{2}(x+6)^2-11\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 5.16.