The minimum value of a quadratic function is the low point at which the function graph has its vertex.

This lesson will define minimum values and give some example problems for finding those values. A quiz will complete the lesson.

## Exercise

The **minimum value of a function** is the place where the graph has a vertex at its lowest point.

In the real world, you can use the minimum value of a quadratic function to determine minimum cost or area. It has practical uses in science, architecture and business.

## How to Determine Minimum Value

There are three methods for determining the minimum value of a quadratic equation.

Each of them can be useful in determining the minimum.The first way is by using a **graph**. You can find the minimum value visually by graphing the equation and finding the minimum point on the graph. The *y*-value of the vertex of the graph will be the minimum. This is especially easy when you have a graphing calculator that can do most of the work for you.

Looking at this graph, you can see that the minimum point of the graph is at *y* = -3.

The second way to find the minimum value comes when you have the equation ** y = ax^2 + bx + c**. If your equation is in the form

*y*=

*ax*^2 +

*bx*+

*c*, you can find the minimum by using the equation min =

*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 minimum point, use the equation to find it.Let’s do an example. Find the minimum point of 3*x*^2 + 12*x* + 2.Since the term with the *x*^2, or ‘a’ term, is positive, you know there will be a minimum point.

To find it, plug the values into the equation min = *c* – *b*^2/4*a*.That gives us min = 2 – 12^2/4(3)This simplifies to *min = 2 – 144 / 12*, which can be further simplified to min = 2 – (12), or min = -10.The third way to find the minimum value is using the equation ** y = a(x – h)^2 + k**.As with the last equation, the

*a*term in this equation must be positive for there to be a minimum. If the

*a*term is positive, the minimum can be found at

*k*. No equation or calculation is necessary; the answer is just

*k*.

Let’s look at an example and find the minimum of the equation (*x* + 13)^2 + 2.Since the *a* term is positive, there will be a minimum at *y* = 2.

## Real World Examples

Now let’s look at some real world examples:The number of bacteria in a refrigerated food is given by the equation *y* = 2*x*^2 + 10. What will be the minimum number of bacteria present?Because the *a* term is positive, we know there will be a minimum for this equation. To find that minimum, we can use the equation min = *c* – *b*^2/4*a*.

We then plug in the numbers from our equation and we get min = 10 – (0^2)/(4 * 2). That simplifies to min = 10 – 0/8 , or min = 10. And that will be the minimum number of bacteria present.Let’s look at another example.

A manufacturer of tennis balls has a daily cost of *y* = 0.01*x*^2 – 0.5*x* + 10. What is the minimum cost for producing tennis balls?Using the equation min = *c* – *b*^2/4*a*, we can find the minimum cost.Once again, we plug in our numbers and get min = 10 – (0.5^2)/(4 * 0.01), which simplifies to min = 10 – 0.

25 / 0.04 = 3.75. So, the minimum cost to produce tennis balls is $3.75.

## Lesson Summary

The **minimum value** of a function is the lowest point of a vertex. If your quadratic equation has a positive *a* term, it will also have a minimum value.

You can find this minimum value by graphing the function or by using one of the two equations.If you have the equation in the form of *y* = *ax*^2 + *bx* + *c*, then you can find the minimum value using the equation min = *c* – *b*^2/4*a*. If you have the equation *y* = *a*(*x* – *h*)^2 + *k* and the *a* term is positive, then the minimum value will be the value of *k*.

Finding the minimum has practical applications in science, engineering and other fields.