Figure 3.1 - The Point P(3,4) Located on a Cartesian Coordinate System
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to graph simple equations in two variables on a rectangular coordinate system.
3.1 - Rectangular Coordinate System - The rectangular coordinate system or the Cartesian coordinate system, is constructed by the perpendicular intersection of two number lines. The horizontal axis is called the abscissa, or the x-axis, and the vertical axis is called the ordinate, or the y-axis. The ordered pair x and y determine points on the plane. (See Figure 3:1).
Figure 3.1 - The Point P(3,4) Located on a Cartesian
Coordinate System
As can be seen in this diagram, the point P(3,4) is determined by the intersection of two lines, one drawn perpendicular to the x-axis (abscissa) at the point representing the number 3, and the second drawn perpendicular to the y-axis (ordinate) at the point representing the number 4. It is important to remember that the first number in an ordered pair represents the x-value, and the second number represents the y-value.
When dealing with equations in two variables, we can construct a picture of them by finding a series of ordered pairs that satisfy the given equation. Connecting these ordered pairs by a smooth curve then gives us the graph of the equation.
Graphing Techniques - Historically, the rectangular coordinate system provided the basis for the development of the branch of mathematics called coordinate geometry. Developed by Rene' Descartes, a 17th century mathematician, coordinate geometry allows us to transform geometric problems into and algebraic setting, and then use the tools of algebra to solve the problem.
Basically there are two kinds of problems in coordinate geometry:
Definition - The graph of an equation in two variables is the set of all points whose coordinates satisfy the equation.
Example 1 - Graph the equation:
3x - y - 6 = 0, -1 < x < 4.
Step 1 - Solve the equation for y:
y = 3x - 6
Step 2 - Substitute the given values of x to find the associated values of y. List in a table.
Table of y vs. x
| x | 3x | -6 | y |
| -1 | -3 | -6 | -9 |
| 0 | 0 | -6 | -6 |
| 1 | 3 | -6 | -3 |
| 2 | 6 | -6 | 0 |
| 3 | 9 | -6 | 3 |
| 4 | 12 | -6 | 6 |
Step 3 - Graph the points on an x-y coordinate system.
Figure 3.2 - The Graph of the Straight Line y = 3x - 6.
As we can see the points for this graph lie on a straight line. The points where the graph intersects the x and y axes respectively are called -
Example 2 - Graph the equation-
x2 - x - y = 2, -2 < x < 2
Step 1 - Solve the equation for y.
y = x2 - x - 2
Step 2 - Substitute the given values of x to find the associated values of y. (List in a table.)
Table of y vs. x
| x | x2 | -x | -2 | y |
| -2 | 4 | 2 | -2 | 4 |
| -1 | 1 | 1 | -2 | 0 |
| 0 | 0 | 0 | -2 | -2 |
| 1 | 1 | -1 | -2 | -2 |
| 2 | 4 | -2 | -2 | 0 |
| 3 | 9 | -3 | -2 | 4 |
Step 3 - Graph the points on an x-y coordinate system.
Figure 3.3 - The graph of the Parabola y = x2 - x -2.
This graph is called a parabola. As you can see this graph also has x and y intercepts.
Board and Class Problem - Graph the equation-
x2 - y = 4, -3 < x < 3
As you can see from this problem, this graph is also a parabola. It is also an example of a graph that is symmetric with respect to the y-axis.
y Axis Symmetry - The graph of an equation is symmetric with respect to the y-axis if replacing x with x results in an equivalent equation.
Example 3 - Graph the equation-
y2 - x + 4= 0 , 4 < x < 8
Step 1 - Solve the equation for y.
Step 2 - Substitute the given values of x to find the associated values of y. (List in a table.)
Table of y vs. x
| x | x - 4 | y |
| 4 | 0 | 0 |
| 5 | 1 | 1 |
| 5 | 1 | -1 |
| 8 | 4 | 2 |
| 8 | 4 | -2 |
Step 3 - Graph the points on an x-y coordinate system.
Figure 3:4 - The Graph of the Parabola y2 - x + 4 = 0.
This graph is also a parabola. It is also an example of a graph that is symmetric with respect to the x-axis.
x- Axis Symmetry - The graph of an equation is symmetric with respect to the x-axis if replacing y with -y results in an equivalent equation.
Example 4 - Graph the equation-
xy = 1, -4 < x < 4
Step 1 - Solve the equation for y.
y = 1/x
Step 2 - Substitute the given values of x to find the associated values of y. (List in a table.)
Table of y vs. x
| x | -4 | -3 | -2 | -1 | 1 | 2 | 3 | 4 |
| y | -1/4 | -1/3 | -1/2 | -1 | 1 | 1/2 | 1/3 | 1/4 |
Step 3 - Graph the points on an x-y coordinate system.
Figure 3:5 - The Graph of the Hyperbola xy = 1
This graph is called a hyperbola, and this particular graph is an example of origin symmetry.
Origin Symmetry - The graph of an equation is symmetric with respect to the origin, if replacing x with -x and y with -y results in an equivalent equation.
Summary of Graphing Techniques -
LESSON #3.2 - The General Objective of this lesson is to study the linear equation in two variables.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to graph the general linear equation in two variables.
3.2 - Linear Equations in Two Variables - In the previous section we saw that the equation
3x - y - 6 = 0
graphed as a straight line. This equation fits the general form
Ax + By = C
which is called the linear equation in two variables, which graphs as a straight line. Therefore, this equation can be graphed by finding two points and connecting them with a straight line.
Example - Graph the equation -
2x - 3y = 6 .
Step 1 - Solve the equation for y.
Step 2 - Determine two points by choosing suitable values for x.
Let x = 0, then y equals -2.
Let x = 3, then y equals 0 .
Step 3 - Locate these points A(0,-2), B(30) and draw a straight line through them.
Board and Class Problems - Page 126/8.
LESSON #3.4 - The General Objective of this lesson is to introduce the concepts of the distance and slope of a straight line.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to:
3.4 - Distance and Slope - Now that we know how to locate points on a rectangular coordinate system, a natural question that arises is: How can we find the distance between two points?
Distance Formula - Consider two points A(x1,y1) and B(x2,y2) as shown in Figure 3:6.
Figure 3.6 - Diagram for the Distance Formula
Since the coordinates of Point C(x2,y1) are determined by A and B it is clear that the lengths a and b are given by
a = y2 - y1 and b = x2 - x1
therefore the distance c is given by the Pythagorean Theorem
Board and Class Problem - Page 141/13.
Slope of a Line - An important property of a straight line is its slope, which is a measure of the steepness of a line. Consider the line joining the two points A and B as shown in the previous diagram. The slope of the line joining the points A and B is defined as the rise divided by the run. The symbol for slope is m, therefore we can write the equation for the slope as -
The slope can assume negative, zero, or positive values. If:
Case 1: m <0, the line goes downhill from left to right.
Case 2: m= 0, the line is horizontal.
Case 3: m> 0, the line goes uphill from left to right.
If the line is vertical, the slope is undefined.
Board and Class Problem - Page 141/56.
LESSON #3.5 - The General Objective of this lesson is to introduce forms for the equation of a straight line.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to:
WORD PROBLEM - None.
3.5 - Determining the Equation of a Line - Recall that there are basically there are two kinds of problems in coordinate geometry:
As we have seen, the general form for a straight line can be expressed as -
Ax + By = C
However, this is not the most convenient form to use. Instead, we will usually work with either the point-slope form, or the slope-intercept form of the straight line.
Point-Slope Form - Since the slope 'm' is defined as -
we can rewrite it as -
y - y1 = m(x - x1)
where the point P2(x2,y2) has been replaced by P(x,y).
Slope-Intercept Form - If we choose P1(x1,y1) to be the point where the straight line crosses the y-axis, then x1 = 0, and if we let y1 be equal to b, the point-slope form yields -
y = mx + b
The form you choose to work with depends upon the information you are given.
Board and Class Problem - Pages 152 to 153/6, 18, 23,50a.
Parallel and Perpendicular Lines - When lines are either parallel or perpendicular to each other, the slopes of each line are related by the following conditions.
The Parallel Condition - Two lines with slopes m1 and m2 are parallel to each other if -
m1 = m2.
The Perpendicular Condition - Two lines with slopes m1 and m2 are parallel to each other if -
m1 x m2 = -1.
Board and Class Problem - Page 141/13.
Graphing: Median Fit Line - Experimentation is crucial to the scientific method. Scientists are constantly measuring quantities and comparing the data they obtain. Therefore the analysis of experimental results for probable experimental error is essential for the proper interpretation of the data collected. It must be noted that some error is always present, when using some measuring instrument, for it is impossible to manufacture parts or instruments with absolute perfection. When a linear relationship exists between the two variables plotted, a best fit line can be drawn. The best fit line should pass as close to all of the points as possible but does not necessarily have to pass through any of them. The best fit line can be represented using the equation for a straight line. There are different ways to determine an equation for a best fit line. The instructions for determining a Median Fit Line are found in the Math 21 Web Pages.
HOMEWORK - Lesson Three Assignment.
COMMENTS -