It is necessary to recall how to graph an equation in two variables such as 2x + 3y = 27. Begin by choosing a pair lines (called axes) which intersect at right angles and call this point of intersection the origin. The horizontal axis is called the x-axis and the vertical axis is called the y-axis. Hence, the plane is divided into four regions called quadrants and each quadrant is named using a roman numeral beginning in the upper right-hand region and continuing in a counterclockwise direction.
The solution set of an open sentence in two variables is the set of ordered pairs (x,y) for which the sentence is true. The sentence 2x + 3y = 27 may be transformed into an equivalent expression.
Hence, the equation has been solved for y in terms of x. Next, arbitrary values of x may be chosen, substituted into the equation, and find the corresponding value of y.
It follows that the points (0,9), (1,25/3), (3,7), (9,3), (-3,11), (-6,13) satisfy the equation 2x + 3y = 27. Indeed, there are infinitely many solutions of {(x,y) | 2x + 3y = 27}.
Begin plotting the points on the coordinate system. The first coordinate is the x-coordinate and the second coordinate is the y-coordinate (also known as abscissa and ordinate, respectively).
Although only two points are required to graph a line, it is good practice to choose a third point as a "test" point.
A relation is defined as a set of ordered pairs of numbers. The set consisting of all first coordinates is the domain; the set of all second coordinates is the range. Hence, 2x + 3y = 27 is a relation whose domain is {0,1,3,9,-3,-6} and whose range is {9,25/3,7,3,11,13}. Only a partial listing of the domain and range may be given since there are infinitely many elements in both. Notice that each element in the domain maps itself to one and only one element in the range. This type of relation is known as a function. Like any other set, a function may be named by a letter such as f, g, or h. Consider the function f(x)=2x. This is read, "f, the function that assigns x to the number 2x." However, f(x) = 2x is the "rule" for the function, not the function itself. The graph of a function is the set of all points (x,y) in a coordinate plane for which the rule of the function assigns y to x.
The most fundamental example of a linear function is f(x) = x. As its name implied, the graph of a linear function is simply a line.
Notice that a one-to-one correspondence exists between each element in the domain with each element in the range.
Consider the following functions:
In the above diagram, the red line is a graph of the function f(x) = x and this function will be called the parent graph from this point on because comparisons of linear functions will be made against this one. The function f(x) = x + 4 is represented by the blue line, f(x) = x + 1 is represented by the green line, f(x) = x - 2 is represented by the gold line and f(x) = x - 3 is represented by the purple line. These lines are all in the same "family" of functions.
In general, if f(x) = x and y =f(x) + c where c is some real number greater than 0, a vertical shift will take place. Specifically, if y = f(x) + c, the graph will shift upward c units; if y = f(x) - c, the graph will shift downward c units. For this reason, f(x) = x + 4 was the same graph as f(x) = x--just shifted 4 units upward; f(x) = x - 3 was the same graph as f(x) = x, but shifted 3 units downward. It is also interesting to note that c is also the y-intercept, which is the point where the graph intersects the y-axis. Since the y-intercept is a point, it may be expressed in the form (0,c).
In addition to creating a table of values and plotting points to draw the graph of a linear function, it is possible to use the steepness of the line, or slope. Consider the function f(x) = 2x - 3 and a few points.
If the two points are chosen, say (2,1) and (3,3), the slope of the line may be found by plotting one of the points and counting the number of units along the y-axis and the number of units along the x-axis necessary to plot the second point.
This technique of counting in order to find the slope may become complicated if the graph is not sketched accurately, or if the first or second point does not consist of integers. For this reason, it is necessary to provide the following formal definition of slope: Let l be a line that is not parallel to the y-axis, and P1(x1,y1) and P2(x2,y2) be distinct points on l. The slope of l is
Since y = f(x) + c is a function whose y-intercept is c and the slope may generate as many points as necessary, the slope and y-intercept together may provide a quick, yet accurate sketch of the graph of any linear function. An equation of the form y=mx + c is said to be in slope-intercept form, where m is the slope and c is the y-intercept. Consider the function
In order to graph this function, it is necessary to first plot the y-intercept. Next, use the slope to locate a second point on the line. It is demonstrated above that the slope may be used to find other points on the line as well.
It should also be noted that if two lines have the same slope, then they are parallel. Consider the functions f(x) = 3x - 4 and g(x) = 3x + 1. It is necessary to note that the function f has a slope of 3 and a y-intercept of -4; the second function has a slope of 3 and a y-intercept of 1. By graphing these two functions, it is visually obvious that these two lines are parallel.
If two lines are perpendicular, then their slopes are negative reciprocals of one another. Consider the functions
The line in red represents the function f and the green line represents g. These two lines intersect and form four 90-degree angles. It may also be generalized that lines with positive slopes begin in the third quadrant and move into the first quadrant (reading the graph from left to right), and lines with negative slopes begin in the second quadrant and move into the fourth quadrant (reading the graph from left to right).
A linear function will graph as a line. The graph may be constructed by either creating a chart of values and plotting points,
or by using the slope and y-intercept.
The function f(x) = x is referred to as the "parent" graph for all linear functions. If y = f(x) + c, the graph moves c units
upward; if y = f(x) - c, the graph moves c units downward.
Parallel lines have the same slope; perpendicular lines have slopes which are negative reciprocals of each other.