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^2/4a.
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 3x^2 + 12x + 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^2/4a.
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 = 2x^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^2/4a. 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 = 0.01x^2 - 0.5x + 10. What is the minimum cost for producing tennis balls?
Using the equation min = c - b^2/4a, 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.
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^2/4a. 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.
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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
1. Find the minimum value of the function using all three methods: looking at the graph, using the formula
2. A business is wanting to minimize their manufacturing costs. It costs
3. Find the minimum value of the function
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
To use the formula
2. Since the cost function is written in the proper form to use the formula
3. This function is written in the form
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