Basic Math and Pre-Algebra

PART 2. Into the Unknown

 

CHAPTER 11. Coordinate Graphing

Graphing Linear Equations

Our coordinate system assigns a pair of numbers to every point and a point to every pair of numbers. The power of such a system is that it lets you give a picture of all the pairs of numbers that solve an equation or inequality with two variables.

Graphs as Pictures of Patterns

The graph of an equation in two variables is a picture of all the pairs of numbers that balance the equation. An equation in two variables has infinitely many solutions, each of which is an ordered pair (x,y). The graph of the equation is a picture of all the possible solutions. Because each of those pairs of numbers fits the same rule, when you plot the points, you find that they fall in a pattern, specifically a line. That’s why these equations are sometimes called linear equations.

Plotting Points

The most straightforward way to graph an equation is to choose several values for x, substitute each value into the equation, and calculate the corresponding values for y. This information can be organized into a table of values. Two points are technically enough to determine a line, but when building a table of values, it is wise to include several more, so that any errors in arithmetic will stand out as deviations from the pattern.

To graph the equation 3x + 2y = 6, make a table with a few possible values of x.

x

3x + 2y = 6

y

(x,y)

-2

3(-2)+ 2y = 6

-6 + 2y = 6

2y = 12

y = 6

6

(-2,6)

-1

3(-1) + 2y = 6

-3 + 2y = 6

2y = 9

y = 4.5

4.5

(-1,4.5)

0

3(0) + 2y = 6

0 + 2y = 6

2y = 6

y = 3

3

(0,3)

1

3(1) + 2y = 6

3 + 2y = 6

2y = 3

y = 1.5

1.5

(1,1.5)

2

3(2) + 2y = 6

6 + 2y = 6

2y = 0

y = 0

0

(2,0)

Once you’ve built your table, plot each of your points. You should see them falling into a line. If a point doesn’t fall in line, check your arithmetic.

WORLDLY WISDOM

If the coefficient of x is a fraction, choose x-values that are divisible by the denominator of the fraction. This will minimize the number of fractional coordinates, which are hard to estimate.

Once you have a line of points, connect them and extend the lines in both directions. Add an arrow to each end to show that the line continues.

When you build a table of values, make a habit of choosing both positive and negative values for x. Of course, you can choose x = 0, too. Usually, you’ll want to keep the x-values near zero so that the numbers you’re working with don’t get too large. If they do, you’ll need to extend your axes, or re-label your scales by twos or fives, or whatever multiple is convenient.

CHECK POINT

Make a table of values and graph each equation.

6. x + y = 7

7. 2x - y = 3

8. y = 3x - 6

9. y = 8 - 2x

10. y = 2/3x - 1

Quick Graphing

Making a table and plotting points will always get you a graph, but it can be a slow process. There are two ways to get the graph quickly, and it’s good to know both, because the way the equation is arranged will determine which method works better.

The first method uses the fact that a point on an axis will always have one coordinate that’s 0. Points on the x-axis have 0 as their y-coordinate, and points on the y-axis have 0 as their x-coordinate. These points on the x-axis and y-axis are called intercepts, and the method is called the intercept-intercept method.

To graph 3x – 4y = 12 by the intercept-intercept method, replace x with 0 and find y. 3(0) – 4y = 12 becomes just -4y = 12, so y = -3. The point (0,-3) is the y-intercept. Go back to the original equation and let y = 0. 3x - 4(0) = 12 becomes just 3x = 12 and x = 4. The x-intercept is (4,0). Plot the x-intercept and the y-intercept, connect them, and extend to make the line 3x – 4y = 12.

Although it can be used for graphing any equation, the intercept-intercept method is best used when the equation has the v and j terms on the same side and the constant on the other side. When the equation is arranged that way, the arithmetic of finding the intercepts is usually simple.

CHECK POINT

Graph each equation by intercept-intercept.

11. x + y =10

12. 6x + 2y =12

13. 2x - 3y = 9

14. x - 2y = 8

15. 6x + 2y = 18

The other quick graphing method uses the j-intercept and a pattern we notice in lines, called the slope of the line. The slope of a line is a measurement of the rate at which the line rises or falls. A rising line has a positive slope, and a falling line has a negative slope. A horizontal line has a slope of zero, and a vertical line has an undefined slope. A line with a slope of 4 is steeper than a line with a slope of 3. A line with a slope of -3 falls more steeply than a line with a slope of -1.

DEFINITION

The slope of a line is a number that compares the rise or fall of a line to its horizontal movement.

The slope of a line is found by counting from one point on the line to another and making a ratio of the up or down motion to the left or right motion. The up or down motion is called the rise, and the left or right motion is called the run. So the slope is

The traditional symbol for the slope is m. If you know two points on the line, you can find the rise by subtracting the y-coordinates and the run by subtracting the x-coordinates. If the points are (x1,y1) and (x2,y2), then

Because you find the slope by subtracting the j’s to find the rise and subtracting the x’s to find the run, the formula for the slope can be written as

The slope of the line through the points (4,-1) and (1,2) is

CHECK POINT

Find the slope of the line that connects the given points.

16. (7,2) and (4,5)

17. (6,-4) and (9,-6)

18. (4,6) and (8,7)

19. (-5,5) and (5,-1)

20. (3,4) and (8,4)

To draw the graph of an equation quickly, arrange the equation so that y is isolated and the x term and constant term are on the other side. This is called y = mx + b form. The value of b is the y-intercept of the line, and the value of m is the slope of the line. In the equation y = 1/2x - 5, the y-intercept is (0,-5) and the slope is 1/2.

Begin by plotting the y-intercept, then count the rise and run and plot another point. Repeat a few times and connect the points to form a line. For the equation y = 1/2x - 5, start by plotting the y-intercept at -5 on the y-axis, then count up 1 (the rise) and 2 to the right (the run), and place a point. Use the slope to plot a few more points by counting up 1 and over 2 a few more times, putting a dot each time. If any points seem out of line, double check your count. Connect the points into a line and extend it in both directions.

CHECK POINT

Graph each equation using y-intercept and slope.

21. v = -3/4x + 1

22. y = -4x + 6

23. y = -3x - 4

24. 2y = 5x - 6

25. y - 6 = 3x + 1

Vertical and Horizontal Lines

Horizontal lines fit the y = mx + b pattern, but since they have a slope of zero, they become y = 0. Whatever value you may choose for x, the y-coordinate will be b.

Vertical lines have undefined slopes, so they cannot fit the y = mx + b pattern, but since every point on a vertical line has the same x-coordinate, they can be represented by an equation of the form x = c, where c is a constant. The value of c is the x-intercept of the line.

CHECK POINT

Graph each equation

26. y = -3

27. x = 2

28. y = 5

29. x = -1

30. y + 1 = 4