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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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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.

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.

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 a*x*^2 + b*x* + c. The a*x*^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.

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.

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.

Riley is working on a new water fountain design. He wants three of the water spouts to shoot water into the air at the same height. The quadratic function for the first two water spouts is f(*x*) = -2*x*^2 + 3*x* + 2. The third water spout has a maximum value of 3.125. Are the three spouts shooting a water parabola of the same maximum value?

Since our problem is in standard form, f(*x*) = a*x*^2 + b*x* + c, Riley can find the maximum value of the equation by first finding the vertex using the formula *x*= -b / 2a. Locate b and a in the original equation and plug those numbers into our formula.

*x*= -3 / 2(-2)*x*= -3 / -4*x*= ¾ or .75

Now, take .75 and plug that back into our original equation to find the *y*-coordinate of the vertex. f(*x*) = -2(.75)^2 + 3(.75) +2. Evaluate the equation. f(*x*) = 3.125. The maximum value of the first two water spouts perfectly matches the maximum value of the third water spout.

So remember, a **parabola** is a u-shaped line that is symmetrical across the line of symmetry. Parabolas that open up or open down have what is referred to as minimum and maximum value. The maximum value of a parabola is the *y*-coordinate of the vertex of a parabola that opens down. The minimum value of a parabola is the *y*-coordinate of the vertex of a parabola that opens up.

We can identify the minimum or maximum value of a parabola by identifying the *y*-coordinate of the vertex. You can use a graph to identify the vertex or you can find the minimum or maximum value algebraically by using the formula *x* = -b / 2a. This formula will give you the *x*-coordinate of the vertex. Simply replace the *x* in your original equation with the value of the *x*-coordinate and then solve for *y*. That will give you the *y*-coordinate of the vertex and either your minimum or maximum value! If the equation is in vertex form, which is *y* = a (*x* - h)^2 + k, just remember that k in that equation is the *y*-coordinate of the vertex.

Finishing this lesson could enable you to reach these goals:

- Give real-life examples of parabolas
- Highlight the characteristics of a parabola
- Locate the minimum or maximum value on the graph of a vertical parabola
- Calculate the minimum or maximum value of a parabola

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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