|Intro to the graphing on the calculator.
Before I give a few tips on using the graphing calculator to check answers and to answer min/max questions, I am going to give a little background on graphs in general and the peculiarities of the graphing calculator. If you would like to skip all of this and just jump immediately to the tips, then click here.
In this class we are going to only cover one and two dimensional graphs in the Cartesian (or rectangular) coordinate system. The one dimensional graph is just the real number line with which you are already pretty familar. Each point on the line corresponds to a real number.
Notice that there is an arrow on only one end. The arrow tells us the direction in which the numbers get larger. In this class we are only going to have numbers get larger as they move to the right on the number line but that is not the case in calculus and physics. It is a good habit to observe where the arrow is located and use it accordingly on your graphs.
The two-dimensional rectagular graph is the result of the cross-product of two sets of real numbers......so the graphical representation is two real number lines placed perpendicular to one another. Again, each number line has just one arrow that indicates the direction that the numbers increase.
In the case of the two-dimensional graph in the rectangular coordinate system, we also label each axis. This is because there is no law that states that the x-axis has to be the horizontal one and the y-axis has to be vertical. The axes could be labeled with totally different letters or the x & y-axes could be switched. To be absolutely clear, always label your axes.
Your calculator has also been hardwired to only accept descriptions of graphs in function form. For your purposes right now, that means that if you want to draw a graph of the line described by
2x + y + 3 = 0,
the you must first solve for y in terms of x----------> y = - 2x - 3
We will talk about functions in a great deal of detail in chapter 4.
What does the graph mean?
Consider the graph of the equation, y = x2 + 4. Exactly what does a point on the graph tell you? Quite a bit, actually. Right now I'll just go through a couple of the simpler ideas and will answer specific questions about the graphs when you ask them.
The point labeled (2,8) says that when x= 2 then y = 8. The ordered pair is describing a relationship between an input to the function, 2, and the output of the function, 8. You can see by looking at the graph that there are an infinite number of such relationships that are being described by the graph since there are an infinite number of points on the graph. For example, the ordered pair (0,4) also describes a point on the graph, which means that when 0 is the input to the function then 4 is the output.
The ordered pair (2,8) also gives us information as to the location of the corresponding point. The point described by (2,8) is 2 units from the y-axis (in the positive direction) and 8 units from the x-axis (in the positive direction). As another example, consider the point (-2, 8). This ordered pair tells you that the point is 2 units from the y-axis (in the negative direction) and 8 units from the x-axis (in the positive direction).
If you have any questions tabout graphs that have been brewing and you would like addressed here (or in an email) the drop me a line.
Often in the course of working a mathematical problem, it is useful to monitor the correctness of your work. For example, you might been asked to complete the square and you would like to check to see if you have done it correctly. A graphing calculator can be useful for doing just that.
Example 1: Use a graphing calculator to see if 8x3 + 27 = (2x + 3) (4x2 - 6x + 9)
(that is, we are checking to see if we factored correctly)
Using a graphing caluator we graph y = 8x3 + 27 and y = (2x + 3) (4x2 - 6x + 9) on the same set of axes. If the two graphs are identical then we can conclude that 8x3 + 27 = (2x + 3) (4x2 - 6x + 9). Notice that the two graphs are indistinguishable from one another so we can assume that our factorization is correct. Note, we cannot say that we have factored completely, but we have no reason to believe that we have made an error thus far.
Example 2: Use a graphing calculator to see if 8x3 + 27 = (2x + 3) (4x2 + 6x + 9)
Since we have already observed in example 1 that 8x3 + 27 = (2x + 3) (4x2 - 6x + 9) and NOT 8x3 + 27 = (2x + 3) (4x2 + 6x + 9), we would not expext the two graphs to be the same. Here is what occurs when we graph y = 8x3 + 27 and y = (2x + 3) (4x2 + 6x + 9) on the same set of axes.
Example 3: When completing the square on x for the function f(x) = 3x2 + 6x - 2, we get
f(x) = 3x2 + 6x - 2 = 3( x2 + 2x ) - 2
= 3 ( x2 + 2x + 1 ) - 2 - 3
Graphing y = 3x2 + 6x - 2 and y = 3 (x + 1) 2 - 5 on the same set of axes we get the following:
Example 4: Use a graph to determine the minimum value of the function f(x) = 3x2 + 6x - 2.
Since the function f is quadratic, we know that its graph will be a parabola. The vertex of the parabola can be read from the graph....it is (- 1, - 5). Since the vertex is the lowest point of the graph, it tells us that the minimum value of the function (output) will be - 5. The output of the function will be - 5 when the input of the function is -1. We can summarize that that sentence by writing the following:
The minimum value of the function, f, is f ( -1) = - 5.
In some problems, will use completing the square (as was illustrated in example 3) to identify the vertex and verify it by looking at the graph.
Example 4: Use a graph to determine the maximum value of the function g(x) = - x4 + 2x - 1.
The graph of the function looks like the following:
So the maximum value of the function, g, is g(.79) = .19
Warning:::::::: It is important to realize that the maximum or mnimum value of the function refers to the OUTPUT of the function so it is just the y value. The point includes both the minimum (or maximum) value as well as the value of the input so THE VERTEX IS NOT the MAX or MIN.
If you are interested in looking at an application that uses the graphing calculator to find a minimum value of a function, read about how to find the point on a curve that is closest to a fixed point.
© 1999 Jo Steig