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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

The maximum value of a function can be found at its highest point, or vertex, on a graph. This lesson will show you how to find the maximum value of a function and give some examples. A quiz will complete the lesson.

The **maximum value of a function** is the place where a function reaches its highest point, or vertex, on a graph. For instance, in this image, the maximum value of the function is *y* equals 5.

Practically, finding the maximum value of a function can be used to determine maximum profit or maximum area. It can be very helpful in running a profitable business or any architecture or building project.

There are three methods for determining the maximum value of a quadratic equation. Each of them can be used in their own unique setting to determine the maximum.

The first way is **graphing**. You can find the maximum value visually by graphing the equation and finding the maximum point on the graph. This is especially easy when you have a graphing calculator that can do the work for you. Again, using this graph, you can see that the maximum point of the graph is at *y* = 5.

The second way to determine the maximum value is using the equation ** y = ax2 + bx + c**.

If your equation is in the form *ax*2 + *bx* + *c*, you can find the maximum by using the equation:

max = *c* - (*b*2 / 4*a*).

The first step is to determine whether your equation gives a maximum or minimum. This can be done by looking at the *x*2 term. If this term is positive, the vertex point will be a minimum. If it is negative, the vertex will be a maximum.

After determining that you actually will have a maximum point, use the equation to find it. For example, let's find the maximum point of:

*-x*2 + 4*x* - 2.

Since the term with the *x*2 is negative, you know there will be a maximum point. To find it, plug the values into the equation:

max = *c* - (*b*2 / 4*a*)

This will give us:

max = -2 - (42 / 4 * (-1))

4 squared is 16, and 4 times -1 is -4. 16 divided by -4 is -4. And, -2 minus -4 becomes -2 plus 4 because two negatives become a positive, so we end up with a maximum value of 2.

There's one more way to determine the maximum value of a function, and that is from the equation:

*y* = *a*(*x* - *h*)2 + *k*

As with the last equation, the *a* term in this equation must be negative for there to be a maximum. If the *a* term is negative, the maximum can be found at *k*. No equation or calculation is necessary - the answer is just *k*.

For example, let's find the maximum of the equation:

-3(*x* - 5)2 - 7

Since the *a* term is -3, there will be a maximum at -7.

Now let's look at a few examples of how this information is helpful in the real world.

Say you have a 250-foot roll of fencing and a large field. You want to construct a rectangular playground area. What is the maximum area your playground could have? Use the equation:

*y* = -*x*2 + 125*x*

Because the a term is negative, we know there will be a maximum for this quadratic equation. To find that maximum, which is the maximum area, we can use the equation:

max = *c* - (*b*2 / 4*a*)

Plugging in our numbers, we get max = 0 - ((1252) / (4 * -1)), or max = -15625 / -4 which becomes max = 3906 feet2. And, that will be the maximum area of our playground using 250 feet of fencing.

Let's look at another one. The height, *h*, in feet that an object is above the ground is given by the equation *h* = -16*t*2 + 64*t* + 190, where *t* is the time in seconds. What is the maximum height of the object?

Using the equationmax = *c* - (*b*2 / 4*a*), we can find the maximum height.

First we plug in our numbers to get max = 190 - ((642) / (4 * -16))

Then we square 64 and multiply the denominator to get max = 190 - (4096 / (-64)), which we can simplify to max = 190 + 64, or max = 254 ft.

The **maximum value** of a function is the place where a function reaches its highest point, or vertex, on a graph. If your quadratic equation has a negative *a* term, it will also have a maximum value. There are three ways to find that maximum, depending on which form of a quadratic you have. If you have the **graph**, or can draw the graph, the maximum is just the *y* value at the vertex of the graph. If you are unable to draw a graph, there are formulas you can use to find the maximum. If you are given the formula ** y = ax2 + bx + c**, then you can find the maximum value using the formula max =

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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

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