LINEAR FUNCTIONS AND THEIR GRAPHS
QUADRATIC FUNCTIONS AND THEIR GRAPHS
EXAMPLES OF EVERYDAY USE
What is a function? According to Edward Begle in his book Introductory Calculus with Analytic Geometry, page 43, a function consists of two things: (1) A collection of numbers, called the "domain of definition"; (2) A rule which assigns to each number in the domain of definition one and only one number. The "range of values" of a function is the collection of all the numbers which the rule of the function assigns to the numbers in the domain of definition of the function.
Letters usually used to denote the "rule" part of the function are f, g, and h.. Though these may be the most used letters, we are not restricted to just these. Letters such as x, y, z, w, and t are typically used to designate values for the domain and range. X is most commonly used for the domain with y used mostly for the range (though no restrictions exist).
A simplistic way of visualizing the concept of functions is illustrated by the following diagram.
So we said that the "function" was the "rule" that was to be used. What do we mean by "the rule"? Let's look at our graph again focusing on the function area.
The function instructs what is to be done with the x-value that has been applied. Let's look at an example.
Example: Let x = 10. The "rule" says to first subtract 2, giving us 8. For the next step we are to divide by 2, giving us 4. Now for the last step we are to take the opposite, which makes our final answer (f(x)) -4. So for any value x that is used, the same steps will be applied. For each x put in, we get a f(x) (the answer).
We normally label f(x) as "y." So we use x to represent the "domain" and y to represent the "range" of the function. The "rule" that is being used is in the form of an "equation." (Note assumption: It is assumed that the student has knowledge of equations) A very simple example of an equation would be: y= x + 8. Think of an equation as the "numbers-and-letters" way of writing a "word" statement.
"One reason for the importance of the concept of function is that functions appear so often, not only in mathematics, but also in other fields. For whenever there is a relationship between two quantities, it defines a function, and before we can study this relationship it is usually necessary to disentangle the function from the verbiage in which it is first presented, the problem being to find as simple an expression for it as possible. There are no fixed rules for doing this." (Begle, 1964
The equation mentioned above is called a polynomial. A polynomial is a function of the form , where coefficients e, d, c, b, and a are real numbers, and "x" is the variable raised to some power (or exponent). A first-degree polynomial, written as , is called a "linear function." The greatest power of "x" is 1. A second-degree polynomial, written as , is called a "quadratic function." Its greatest power of "x" is 2. We will examine both of these in-depth later.
Let's look at examples from the "real" world where functions are used (even though the person using it may not even know it).
Example: A painter wants to know how much paint he would need to paint a rectangular box having a square base. For this he needs to know the total surface area of the box. He knows the volume of the box is 10 cubic feet. What function could he use to find the surface area of the box in terms of the side of the base?
Let's have "x" to represent one side of the base of the box. The area for the squared base then would be A= length times width. Since it is a square, the length and width will be equal. Therefore, the area of the base would be x^2.
The formula for volume is V= length times width times height. So from this we get height= 10/x^2. Each side of the box would have area x(10)/x^2 = 10/x. Remember that there are four sides and a top and a bottom. Therefore we have the total surface area represented by 40/x + 2x^2. The function derived from this would be f(x)= 40/x + 2x^2, with x > 0. Using this equation (function), the painter could easily find the total surfaces of other boxed with this shape but with different values for x. With each new x used, a new f(x), or y, would be gotten.
Example: A toy manufacturer spends 200 dollars each week for the rent and upkeep of his plant. He has figured out that it costs him 10 dollars to produce one toy. Selling the toy for "p" dollars, he can sell 200 - p each week. He needs to find the function that would show his weekly profit as a function of the selling price.
Assuming p>0, his income would be (200 - p)p dollars, and his total cost is 200 + 10(200 - p). Therefore, his profit would be (210)p - p(62) - 2200. Since p was assumed to be greater than zero, it also must be less than or equal to 200. The domain for this problem could be represented by (0, 200]. So the function the toy manufacturer needs to use would be f(p) = (210)p - p^2 - 2200. Using this equation each week, he could put in that week's p-value and be able to see what his profit was for that particular week.
Example: A carpenter needs to install a window for a new house that is being built. The window has the shape of a rectangle with a semicircle on the top. He wants to know the area that the window will take up. He needs a function (an equation) that would relate the area of the window to the length of the base of the rectangle.
He knows that the area of a rectangle is equal to the length times the width. He also knows that the area of a semicircle is equal to 1/2(pi times the radius squared). The radius of the semicircle is 1/2 the length of the base of the rectangle. In effect, he will be adding the area of the rectangle to the area of the semicircle. If he varies the length of the rectangle, it will directly vary the area of the semicircle. Therefore, an equation that he could use would be
So we can see that functions really are used outside of the math classroom. Now we will examine different kinds of functions and their graphs.
The first function we will study is the "linear" function, a first- degree polynomial having the basic equation . The basic graph that should come to mind for this function is a "straight line." From now on we will let "y" represent "f(x)" in working with our equations. So we have . Let's graph a few examples of linear functions.
(1) For our first example, we will use the equation "y= 2x +3." As we put in various values for x, we will get a corresponding value for y.
Looking at the table we can see that the value for y is 6 when x= 1. When x= 0, the value for y is 3. When x= -1, y is then 1. Now let's represent this equation with the given values using a graph.
Let's do another graph with different values for "a" and "b" and see what happens to the graph. We will use the same values for "x." The equation that we will use is .
So using the same "x"-values but with different values for "a" and "b", still gave us a straight line but in a different position on the graph. Now let's graph the simple equation .
This is a graphing of a very simple equation. Whatever value x equals to, y also equals to the same value. Now let's make a slight change to the equation and see what effect, if any, it can have on our graph. Now our equation is , with x = y = 1 and the variable "n" will exist in the interval [-10, 10]. Let's see how varying "n" will affect the graph.
Yes, that seemingly "small" change to the equation does make a big difference.
Now let's graph two equations on the same graph, and
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We see in this graph that the two lines intersect each other. The place where the intersection occurs is a "point." In this case, the point of intersection is (1,1). What happens if the graph of two linear equations has no points in common? Let's see. We will use the two equations: and .
Without having any points in common, the two straight lines will not intersect, meaning there will be no point of intersection. In this case the lines are called "parallel lines." When we think about graphs we usually think of the type above in the x-y plane. What would happen if we needed to work in a three dimensional plane where we have "x", "y", and "z" as the variables? Is a linear equation even possible in 3-D?
First let's look at what an equation would look like for a three-space graph. In the two-space linear equation we only had to work with two variables, "x" and "y" (or whatever labels you choose). It follows that working in three-space would deal with three variables, say "x", "y", and "z." So for our general linear equation in three-space, we will use "." First we will look at a basic three-space graph. We will have a,b, and c to be equal to 2, and the x, y, and z are equal to 1. You will see in this graph that instead of a line, the result is a "plane." Click on the "click here" below and investigate a plane revolving in its three-space area.
That is definitely different from our straight line graph. Now let's graph two 3-D equations and see what will happen.
After graphing two equations, and , we see two planes that intersect each other. Compare this intersection with the intersection of the two straight lines above. (click here) What is the major difference between the two intersections? In the 2-D graph the intersection is a point. In the 3-D graph the intersection is a line, not just one point. So what happens if in 3-D the planes do not have any common points? We will use the equations and .
As with the parallel lines above, the two planes are parallel, also. Therefore the two planes will not intersect each other.
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Now we will examine the quadratic functions and graphs. As mentioned above the equation form for the quadratic is . Notice that the greatest power of "x" is 2, that is, x-squared. First we will look at a simple quadratic graph. We will graph with "a" = 1, and both "b" and "c" are equal to zero.
Does changing the values for a,b, and c have any effect on the graph? Let's see. First we will set a= 2, b= 2, and c= 1.
Let's see what happens when "a" is equal to a negative number. Let a= -1 while keeping the rest the same.
So having a negative value for "a" really has an effect on the graph. We see that when the value for a is positive, the graph "cups" upward. When a is negative, the graph "cups" downward. So we see that each part of the quadratic equation has its own particular way of affecting its graph.
As with the linear equation, let's look at the quadratic equation in the 3-dimension. That should be very interesting. First we will graph one of the simpler quadratics taken into the third dimension. We will graph the following equation: , having c= 0 and a, b, both = 1.
WOW. Now that is definitely different! Go back and compare this with the graph in 2-dimension. Quite a bit different, don't you think? Click Here to see the above graph in motion. It's really neat!
Let's experiment further with the quadratic function in 3-D by varying the coefficients values. We will be using the above equation during our explorations. First let's see what happens when we set a = 1, b= 1, and c= 1. When c equals a value other than zero, the parabola moves from the (0, 0, 0) point. Because of this, I will have c equal to zero for the rest of the experiments just for the sake of room on the graph. It will make viewing the graph easier. Click here for the movie of the graph below.
For our next graph let's set a= 2 and keep b= 1.
What differences do you see between the above two graphs? Since c= 0 the graph is positioned as we wanted. What happened with respect to the sides of the parabola? The "a" coefficient caused opposite sides to increase, while the sides regulated by the "b" coefficient remained the same. Click here for the movie.
Now we will have both "a" and "b" equal to 2. Can you guess as to what will happen? Let's see if you are correct.
It may be hard to effectively see the graph. Viewing the movie will illustrate the changes to the graph better. But you will see that having both "a" and "b" equal to 2 caused all sides of the parabola to increase. Click here for the movie.
Let's see what will happen if a= 1/2 and b= 2. Notice any changes?
That 1/2-coefficient really did affect the sides of the parabola. For a better view Click here for the movie. As the graph rotates in the movie, you will be better able to view the sides. Having this result, what do you think will happen if both "a" and "b" are equal to 1/2?
The 1/2-coefficient greatly reduced to height of the sides. Click here for movie. Now let's have a= -1/2 and b= 1. What do you think will happen?
Well, I did not expect this to happen. It certainly is interesting, though. Click here for movie. Let's try again with both "a" and "b" equal to -1/2.
So we see that if both coefficients are negative values, the parabola turns down. Interesting! And neat looking! Click here for movie.
Begle, Edward (1964). Introductory Calculus with Analytic Geometry. Holt, Rinehart, and Winston, Inc. U.S.A.
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