The profit on your school fundraiser is represented by the quadratic expression
is your price point. What price point will result in a maximum profit and what is that profit?
So far, we have only used the
of a quadratic equation,
to graph a parabola. From standard form, we can find the vertex and either factor or use the Quadratic Formula to find the
of a quadratic equation is
is the same value as in standard form, and
intercepts. This form looks very similar to a factored quadratic equation.
to intercept form and find the vertex. Graph the parabola.
First, let’s change this equation into intercept form by factoring.
and the factors of 20 that add up to 9 are 4 and 5. Expand the
Notice, this does not exactly look like the definition. The factors cannot have a number in front of
. Pull out the 2 from the first factor to get
. Now, find the vertex. Recall that all parabolas are symmetrical. This means that the axis of symmetry is
intercepts or their average.
This is also the
coordinate of the vertex. To find the
coordinate, plug the
value into either form of the quadratic equation. We will use Intercept form.
The vertex is
. Plot the
intercepts and the vertex to graph.
The last form is vertex form.
is the vertex and
is the same is in the other two forms. Notice that
is negative in the equation, but positive when written in coordinates of the vertex.
Find the vertex of
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
. -3 is 4 units away from 1 (the
coordinate of the vertex). 4 units on the
side of 1 is 5. Therefore, the
coordinate will be 11. Plot (1, 3), (-3, 11), and (5, 11) to graph the parabola.
into vertex form.
To change an equation from standard form into vertex form, you must complete the square. Review the
Completing the Square
Lesson if needed. The major difference is that you will not need to solve this equation.
To solve an equation in vertex form, set
and solve for
Intro Problem Revisit
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.
From this we can see that the x-intercepts are 40 and 40. The average of 40 and 40 is 40 we plug 40 into our original equation.
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!
1. Find the intercepts of
and change it to standard form.
2. Find the vertex of
and change it to standard form.
to intercept form and graph.
to vertex form and graph.
1. 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
2. The vertex is (-4, -5). To change the equation into standard form, FOIL
, and then subtract 5.
3. To change
into intercept form, factor the equation. The factors of 45 that add up to 18 are 15 and 3. Intercept form would be
. The intercepts are (-15, 0) and (-3, 0). The
coordinate of the vertex is halfway between -15 and -3, or -9. The
coordinate of the vertex is
. Here is the graph:
4. To change
into vertex form, complete the square.
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
intercepts are (7, 0) and (-1, 0).
is the vertex.
Fill in the table below. Either describe how to find each entry or use a formula.
Find the vertex and
intercepts of each function below. Then, graph the function. If a function does not have any
intercepts, use the symmetry property of parabolas to find points on the graph.
Change the following equations to intercept form.
Change the following equations to vertex form.
Change the following equations to standard form.