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Using Quadratic Models to Find Minimum & Maximum Values: Definition, Steps & Example

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  • 1:01 Quadratic Functions
  • 2:46 Values Graphically
  • 4:53 Values Algebraically
  • 8:15 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values algebraically and graphically.

Parabolas in Real-Life

Riley and Sam are brothers. Riley works as a landscape architect designing giant water fountains for big hotels and businesses. Riley has to do a lot of math to calculate the size, shape and direction of the water that is flowing out of the water fountain. His brother Sam is a tennis coach for a local high school. Sam reminds his students that it is important to bounce the tennis ball one time on the opponent's side of the net. The flow of the water out of a fountain and the bounce of a tennis ball on a court both create a similar u-shape called a parabola.

Parabolas are found in many areas of life. Bridges create perfect examples of parabolas. A ball being thrown through the air or bounced off of the ground makes a parabola shape during its journey. The tallest hill on a roller coaster will often create the shape of a parabola.

Parabolas and Quadratic Functions

So what is a parabola? A parabola is a u-shaped line that is symmetrical across the line of symmetry, which means if you draw a line exactly down the center of the u-shape, it creates a mirror image across that line. This u-shape can open up and create a genuine u, or it can open down, left, or right.

The very top or the very bottom of that u-shape where the line goes through the center of the parabola is called the vertex. The vertex of a parabola is the peak of the curve of the parabola. Again, depending on if the parabola is opening up or opening down, the vertex might either be at the very top or the very bottom of that curve.

Mathematically, a parabola is created with a quadratic function, which is an equation with the form of ax^2 + bx + c. The ax^2 part is the part of the function that creates that nice rounded curve on that u shape rather than a v.

Parabolas that open up or open down, like this, have what is referred to as a minimum and a maximum value.

examples of vertical parabolas
maximum valueyminimum valuey

Finding Values Graphically

Let's think about the meaning of the words 'minimum' and 'maximum.' 'Maximum' usually means 'most' or 'more' or 'largest.' If I got the maximum number of points on a test, then I know I got the most points. If Susan got the maximum amount of fries, we know she has the largest amount of fries she could get. 'Minimum' usually means 'least' or 'less' or 'smallest.' So when I think of the minimum amount of points I have to score to win a game, I'm probably thinking of the smallest number.

The same concept works with minimum and maximum values of a parabola. If the parabola opens down, which means it makes a hill shape, then the very top of that hill would be the maximum value of the parabola. It would be the maximum height a person could reach if he or she were climbing that hill. If the parabola opens up and creates a valley, or u-shape, then the very bottom of that valley would be the minimum value. It would be the very lowest point a person could travel in that valley.

We can identify the minimum or maximum value of a parabola by identifying the y-coordinate of the vertex. Take a look at this graph. The vertex is located at the point (2.5, -.5), and the parabola opens up. That means that the parabola has a minimum value, which is y = 2.5.

upward facing parabola

What about this graph? The vertex is located at the point (1,6), and the parabola opens down. That means that the parabola has a maximum value, which is y = 6.

downward facing parabola

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