3.07 Graphing lines
Introduction
We have learned how to make tables of values of proportional relationships and functions. We can also write the slope intercept form of the equation of a line. It's now time for us to graph lines using the slope-intercept form of the equation of the line and using tables of values.
Graph from tables and equations
Linear functions have graphs that are straight lines. These functions can also be represented by tables of values or equations.
A table of values can be graphed on the coordinate plane by plotting the ordered pair (x,y) from the table.
A linear equation in slope-intercept form, expressed as y=mx+b can be plotted on the coordinate plane. The slope m and the y-intercept, b can be used to graph any line. When we are given an equation in slope-intercept form, we are given one point (the y-intercept) and the ability to find a second point (using the slope), so we are all set.
- Start by plotting the y-intercept from the equation
- Identify the rise and run from m in the equation. If the slope is given as an integer, remember that we can write \dfrac{\text{rise}}{\text{run}}=\dfrac{m}{1}.
- Put your pencil on the y-intercept and count "rise" squares up or down and then "run" squares to the right. Repeat as needed
- Connect the two or more points to form a line, extending beyond the two points.
Here is a little more detail on step 2.
Examples
Example 1
Consider the equation y = \dfrac{x}{2} + 5
A table of values is shown below:
| x | -4 | -2 | 0 | 2 |
|---|---|---|---|---|
| y | 3 | 4 | 5 | 6 |
Plot the points in the table of values on the coordinate plane.
Is the graph of y = \dfrac{x}{2} + 5 linear?
Example 2
Graph the line y=3x+2 using its slope and y-intercept.
A linear equation is said to be in slope-intercept form when it is expressed as
To graph any line we only need two points that are on the line. When we are given an equation in slope-intercept form, we can use the y-intercept as the first point. Using the slope, we can find the second point by counting up and accross the y-intercept.
Horizontal and vertical lines
Recall that lines can also be either horizontal or vertical. These types of lines will not look like slope-intercept form. However, they do follow another type of special pattern.
Examples
Example 3
Plot the line x=-8 on the coordinate plane.
Horizontal lines are the set of all points with a fixed y-value. They are parallel to the x-axis and have equations of the form y=a, where a is a real number.
Vertical lines are the set of all points with a fixed x value. They are parallel to the y-axis and have equations of the form x=a, where a is a real number.
Non-linear graphs
Not all relationships between two quantities form a linear graph on the coordinate plane.
The following are examples of non-linear graphs and their corresponding equations.
What can we notice about the equations of non-linear graphs?
Linear functions are functions that appear as straight lines when they are graphed. The equation for a linear function can be represented by y=mx+b. Non-linear functions as the name suggests, is a function whose graph is NOT a straight line. Its graph can be any curve other than a straight line. Non-linear functions are represented by equations that cannot be represented by y=mx+b or y=mx.
Examples
Example 4
Which of the following graphs are non-linear graphs?
Example 5
Identify two equations representing non-linear functions.
Non-linear functions have graphs that are NOT straight lines.
Non-linear functions have equations NOT in the form of y=mx or y=mx+b.