# Minimum Values: Definition & Concept

## 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.

To unlock this lesson you must be a Study.com Member.

Create your account

## Practice Problems - Finding Minimum Values

In the following practice problems, students will find the minimum value of a quadratic function by using a graph, using the formula

*k*when the function is written in the form

## Practice Problems

1. Find the minimum value of the function using all three methods: looking at the graph, using the formula

*k*from the form

2. A business is wanting to minimize their manufacturing costs. It costs

*x*thousand units of the commodity. What is the minimum cost for production?

3. Find the minimum value of the function

## Solutions

1. The minimum value is the *y* value of the lowest point on the graph. Looking at the graph, this is -1.5.

The function is given in the form

*k*= -1.5, so using this method agrees with the graph method.

To use the formula

*a*,

*b*, and

*c*. We have

*a*= 2 ,

*b*= -12,

*c*= 16.5. Using the formula, we find that the minimum is

2. Since the cost function is written in the proper form to use the formula

*a*= 5,

*b*= -20, and

*c*= 150 and so

3. This function is written in the form

*k*= 13.

### Register to view this lesson

### Unlock Your Education

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.

Become a MemberAlready a member? Log In

Back