Jennifer has an MS in Chemistry and a BS in Biological Sciences.
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.
What Is Maximum Value?
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.
How to Determine Maximum Value
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 ax2 + bx + c, you can find the maximum by using the equation:
max = c - (b2 / 4a).
The first step is to determine whether your equation gives a maximum or minimum. This can be done by looking at the x2 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:
-x2 + 4x - 2.
Since the term with the x2 is negative, you know there will be a maximum point. To find it, plug the values into the equation:
max = c - (b2 / 4a)
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.
Real World Examples
Now let's look at a few examples of how this information is helpful in the real world.
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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 = -x2 + 125x
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 - (b2 / 4a)
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 = -16t2 + 64t + 190, where t is the time in seconds. What is the maximum height of the object?
Using the equationmax = c - (b2 / 4a), 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 = c - (b2 / 4a). If you have the equation y = a(x-h)2 + k and the a term is negative, then the maximum value is k.
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