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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Quadratic functions are very useful in the real world. Watch this video lesson and learn how these functions are used to model real world events such as a falling ball.

In this video lesson, we will talk about how **quadratic functions**, the function of a degree of 2, are used in the real world to model real-world scenarios. Remember that a function with a degree of 2 has, at most, three terms with the highest exponent of 2. All of our quadratic functions follow the standard form of *f*(*x*) = *ax*^2 + *bx* + *c*, where *a*, *b*, and *c* are your coefficients and *x* is your variable.

When used to model real-world scenarios, though, your variable can be another letter; it is not always *x*. It can be *t* to stand for time, or *h* to stand for height, or any other letter to stand for whatever is needed by the problem. For example, this quadratic function is the basic function for an object that is either thrown up into the air or just let go:

The *g* comes from the force of gravity. It is either 4.9 in meters or 16 in feet. The *v* sub 0 stands for the initial velocity of the object, and *h* sub 0 is the height from which the object is thrown up or dropped. Notice the exponent of 2.

Our formulas for calculating area are also quadratic functions. For example, the function to calculate the area of a square is *A* = *s*^2, where *A* stands for the area and *s* stands for the measurement of a side of the square. Do you see the exponent of 2?

What kinds of real-world problems can we expect to solve using these quadratic functions? We can solve problems where we want to find how much time a certain object stays in the air before hitting the ground after being thrown into the air, or dropped, or thrown down from a certain height.

For instance, say someone was throwing candies from the third floor. You are on the bottom and want to know how long you need to wait before putting your bucket out to catch the candies. You can use a quadratic function to find out.

And we can also solve problems like finding the best dimensions for a yard to hold the largest area given only a certain amount of fencing. Say you were given only one roll of fencing. You can use a quadratic function to help you find the largest area you can enclose with that amount of fencing. You want to see how this works with a couple of examples?

Okay! Let's talk about the candy one first. This deals with gravity, so our function is based on how objects fall when under the force of gravity. Living on this planet Earth, you know that everything is affected by the force of gravity, which happens to be the same for all objects. Therefore, we have this formula for objects falling under the force of gravity:

The little *g*, which is derived from the mathematical value of gravity, is 4.9 if we are dealing with meters and 16 if we are dealing with feet. Here is our problem:

'The candy man is throwing candies out the third floor window of his candy factory. He throws the candies up into the air at a velocity of 16 feet per second. The height of the third floor window is 32 feet. How many seconds does it take for the candies to hit the ground?'

We see that this problem gives us our *v* sub 0, our initial velocity, as well as *h* sub 0, our initial height. Since we are dealing with feet, our *g* is 16. Plugging in our values of *g* = 16, *v* sub 0 = 16, and *h* sub 0 = 32 into our gravity function, we get this formula:

This function gives us the height of our object, our candies, at any given point in time. To find when our candies will hit the ground, we set this function equal to 0 for a height of 0:

We then use algebra skills to solve this quadratic function. We first divide by -16. We get 0 = *t*^2 - *t* - 2. We can then solve by factoring. It becomes 0 = (*t* - 2)(*t* + 1).

We get answers of *t* = 2 and *t* = -1. Which one of these answers makes sense for us? Well, we can't go backwards in time, so *t* = 2 is the answer that makes sense. It takes 2 seconds for the candies to hit the ground.

Our next example takes a look at **optimization**, or making the most of a situation. Our problem is this:

'We have 500 feet of fencing available for us to use. What are the dimensions of the largest size rectangular yard this amount of fencing can make?'

To solve this problem, we need to make use of the area formula for a rectangle. The formula is *A* = *L* * *w*. We have two variables. To turn this into a quadratic function, we need another equation that we can solve for one of the variables to plug it into the area formula.

This second equation we will get from our 500 feet of fencing. Our formula for our perimeter is 2*L* + 2*w* = 500. Solving this *w*, we get *w* = 250 - *L*. Plugging this into the formula for area, we get *A* = *L* * (250 - *L*) = 250*L* - *L*^2 = -*L*^2 + 250*L*.

If we were to graph this function out, we would get a parabola. The largest area this function can hold is then found by finding where the tip or vertex of our parabola is. We can find the length by calculating -*b*/2*a*, which gives you the *h* of the vertex (*h*, *k*). The *h* corresponds to the length in our area calculation.

Calculating -*b*/2*a*, we get -250/2(-1) = 125. So, our length is 125. To find our width, we will use *w* = 250 - *L* and plug in our *L.* We get *w* = 250 - 125 = 125. Ah, so our width is also 125. That means that our rectangle is also a square.

So a rectangular yard that measures 125 by 125 feet is the largest yard we can make. If we plugged in 125 into our area formula, we would find the area of our largest yard.

Let's review what we've learned. We learned that **quadratic functions**, the functions with a degree of 2, are used in the real world to model different scenarios. The standard form of all quadratic functions is *f*(*x*) = *ax*^2 + *bx* + *c*, where *a*, *b*, and *c* are your coefficients and *x* is your variable. Quadratic functions are used to model gravity problems where we want to find how long it takes an object to fall back to the ground after it's been thrown into the air, dropped from a certain height, or thrown down from a certain height.

Quadratic functions are also used to help solve **optimization** problems, or problems that want you to find the largest area a certain amount of fencing can hold. To solve these problems requires algebra skills once we have our formula. And these problems do take some thinking and common sense to set up properly and solve. For example, when the problem asks about the time it takes for an object to hit the ground, you need to think and tell yourself that the ground is the same as a height of zero.

You should have the ability to do the following after this lesson:

- Define quadratic functions
- Identify the standard form for quadratic functions
- Explain how to use quadratic functions to solve gravity and optimization problems

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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- How to Solve a Quadratic Equation by Factoring 7:53
- How to Solve Quadratics with Complex Numbers as the Solution 5:59
- Graphing & Solving Quadratic Inequalities: Examples & Process 6:14
- Graphing a System of Quadratic Inequalities: Examples & Process 8:52
- Applying Quadratic Functions to Motion Under Gravity & Simple Optimization Problems 7:42
- Go to Working with Quadratic Functions

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