MATH 101: GRAPH AND WRITE EQUATIONS AND INEQUALITIES/Graphing Lines

Graphing Points and Lines

The coordinate system, Cartesian plane, consists of two perpendicular lines labeled as the x and y-axis. An ordered pair, such as (2, 3), is graphed going 2 spots to the right on the x-axis and 3 spots up on the y-axis. There are 4 quadrants in the Cartesian plane. A quick way to remember them is that they go in order of a ‘C’ starting in the upper right quadrant and going counter-clockwise to the lower right quadrant.

The slope of a line is the rise over run or change in y over change in x.

m= y2-y1

x2-x1

Basically, pick a point to start with and set up the slope formula. The y’s go on the top of the fraction. Subtract the y’s from one another. The x from the first point you picked will be the first x on the bottom. Then subtract the x’s. Reduce the fraction to lowest terms. Leave the answer as a fraction, not decimal or mixed number. You will need it to graph.

There are 4 types of slopes.

Positive- goes from quadrant 1 to 3

Negative- goes from quadrant 2 to 4

Zero- this is a horizontal line <---->

Undefined- this is a vertical line

Example: Find the slope of (1, 3) (5, 6).

6-3 = 3

5-1 4

Slope-intercept form: y=mx+b where the m= slope and b= y-intercept

Standard form: Ax+By=C

Point slope form: y-y1= m(x-x1) where m= slope and (x1, y1) is a point on the line

x-intercept: the point where the line crosses the x-axis

To find it, let y=0 and solve for x.

y-intercept: the point where the line crosses the y-axis

To find it, let x=0 and solve for y.

Equations and inequalities can be graphed using 3 different methods: slope and y-intercept, table of values, and using the intercepts.

1. Slope and y-intercept

a. Identify the slope and y-intercept. To graph, plot the y-intercept and then the slope using rise over run.

b. If the equation is not in the slope-intercept form, convert it to this form to easily identify the slope and y-intercept.

2. Table of values

a. Use a table of values by choosing 3-5 possible values for the first variable. Plug each one into the equation and solve for the other variable. Graph the points.

3. Intercepts

a. Find both the x and y intercepts of the equation. Graph and connect the points.

Horizontal and Vertical Lines

A horizontal line goes left to right (like the horizon) and is represented as an equation using y=b. The slope of a horizontal line is 0.

A vertical line goes up and down and is represented as an equation using x=a. The slope of a vertical line is undefined.

Examples:

Writing the equations of special lines is simple. For vertical lines, use the x-coordinate and substitute it in for a in x=a. For horizontal lines, use the y-coordinate and substitute it in for b in y=b.

Examples:

Write the equation of a vertical line passing through the point (4, 7).

x=4

Write the equation of a horizontal line passing through the point (4, 7)

y=7

Graphing

You will graph only from slope-intercept form. If it is in another form, you will need to convert the equation into slope-intercept. Watch the signs! Once it is in slope-intercept form, start with the ‘b’ or the y-intercept. Graph the y-intercept on the y-axis. Then use the slope, ‘m’ to plot the slope. Remember that the slope is rise/run. So first you move up or down and then left to right. Up and right are for positive (+) numbers. Down and left are for negative (-) numbers. You will see a ‘stair’ pattern being made. Once you have a few points plotted, connect the dots.

Example:

y=2x-1

Start by graphing (0, -1) on the y-axis.

Plot the slope, 2/1.

Go up 2, right 1, dot, up 2, right 1, dot.

To extend the line the other way, reverse the signs, -2/-1 (it is still the same)

Go down 2, left 1, dot, down 2, left 1, dot.

This will extend the graph in both directions.

The lines we are graphing are linear, meaning ‘line’. So, they should be straight diagonal lines, no zig zags. The only exceptions are the two special lines of x=? and y=?, but those are straight too. x=a is vertical and y=b is horizontal.

Linear Inequalities

Solve inequalities the same as equations with a few exceptions. When you multiply or divide by a negative number, flip the inequality sign. They are graphed slightly differently. Graph as you would an equation, using the y-intercept and slope. But to connect the lines, it will be either a ---- dotted or _____ solid line. The signs < and > mean that the line is dotted. The signs < and > mean that the line is solid.

There is one more thing to know about graphing inequalities. You will also shade a side. There are two ways to do this. If the equation is in slope-intercept form, the > (greater than) means that the shading will be above the line. If the equation is in slope intercept form, the < (less than) means that the shading will be below the line. The second method is to test a point. Pick any point on the graph, but the point cannot be on the line. (0, 0) is a good choice, if not on the line, and plug it into the inequality. If the side with the point makes the problem ‘true’, shade that side. If the point makes the inequality ‘false’, shade the other side.