11: Graphs

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Which cyclist will win the race? What will the winning time be? How many seconds will separate the winner from the runner-up? One way to summarize the information from the race is by creating a graph. In this chapter, we will discuss the basic concepts of graphing. The applications of graphing go far beyond races. They are used to present information in almost every field, including healthcare, business, and entertainment.

• 11.1: Use the Rectangular Coordinate System (Part 1)
Just as maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system.  In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point.
• 11.2: Use the Rectangular Coordinate System (Part 2)
Equations with two variables can be written in the general form Ax + By = C. An equation of this form is called a linear equation in two variables. Linear equations in two variables have infinitely many solutions. For every number that is substituted for x, there is a corresponding y value. This pair of values is a solution to the linear equation and is represented by the ordered pair (x, y).
• 11.3: Graphing Linear Equations (Part 1)
The graph of a linear equation Ax + By = C is a straight line. Every point on the line is a solution of the equation. Every solution of this equation is a point on this line. The method we used at the start of this section to graph a linear equation is called plotting points, or the Point-Plotting Method. You can use two points to graph a line, but if you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work.
• 11.4: Graphing Linear Equations (Part 2)
In this section, we will graph equations with only one variable. That is, there is just x and no y, or just y without an x. A vertical line is the graph of an equation that can be written in the form x = a. The line passes through the x -axis at (a, 0). A horizontal line is the graph of an equation that can be written in the form y = b. The line passes through the y-axis at (0, b).
• 11.5: Graphing with Intercepts (Part 1)
Every linear equation has a unique line that represents all the solutions of the equation. At first glance, two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. To graph a linear equation by plotting points, you can use the intercepts as two of your three points.
• 11.6: Graphing with Intercepts (Part 2)
We can use the form of equation to choose the most convenient method to graph its line.  If the equation has only one variable, it is a vertical or horizontal line. If y is isolated on one side of the equation, graph by plotting points. Choose any three values for x and then solve for the corresponding y- values. If the equation is of the form Ax + By = C, find the intercepts. Find the x- and y- intercepts and then a third point.
• 11.7: Understand Slope of a Line (Part 1)
The steepness of the slant of a line is called the slope of the line. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. The slope formula states that slope is equal to y of the second point minus y of the first point over x of the second point minus x of the first point.
• 11.8: Understand Slope of a Line (Part 2)
In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines. Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.
• 11.9: Graphs (Exercises)
• 11.10: Graphs (Summary)

Figure 11.1 - Cyclists speed toward the finish line. (credit: ewan traveler, Flickr)

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