Hi there. In this math post I cover graphing linear equations.
Topics
- Slope & y-intercept Review
- Graphing Linear Equations With Table Of Values
- Phone Plan Example
- Closing Notes
Slope & y-intercept Review
The equation of a line follows the format of y = mx + b. The value m refers to the slope of a line, the value b is the y-intercept, the independent variable x is an input number and the dependent variable y is the output from mx + b.
y=mx + b
Note that the y-intercept is the y-value when x = 0.
Graphing Linear Equations With Table Of Values
In this section I showcase graphing linear equations with a table of values.
Example One
Graph y = 3x.
Use a table of values. I generally use small and easy values for x such as 0, 1, 2, -1 and -2. Use each x-value as an input to get the corresponding y-value.
When x = 0 then y = 0. If x = 1 then y = 3. When x = -2 then the output y becomes negative six (from -2 times 3). I go with five points but two points is enough the create a line.
| x | y |
|---|---|
| -2 | -6 |
| -1 | -3 |
| 0 | 0 |
| 1 | 3 |
| 2 | 6 |
You have the five points of (-2, -6), (-1, -3), (0, 0), (1, 3) and (2, 6). Plot these points on a co-ordinate grid. Then you can connect the dots and create a line.
For the line I use Desmos and a screenshot tool to show this image.
Example Two
Graph y = 5x - 2.
This linear equation is slightly more involved. Be careful of negative numbers in general. I use the x-values -1, 0 and 1.
| x | y |
|---|---|
| -1 | -7 |
| 0 | -2 |
| 1 | 3 |
The three points are (-1, -7), (0, -2) and (1, 3). Notice that the y-value when x = 0 is negative 2. This negative 2 is the same -2 from the linear equation. This is the y-intercept.
Using Y-Intercept and Slope To Graph
The table of values method of graphing a line can take some time. A different method that could be faster for some is using the y-intercept and slope from the linear equation.
The y-intercept is from the number only part of the y = mx + b equation of a line. Then you can use the slope to plot the next point(s). This will make more sense through examples.
Example One
Graph y = -2x + 1.
The y-intercept can be obtained from the 1. It is the point (0, 1). Graph this point first.
From y = -2x + 1, the slope is negative two. To get the next point you can do down 2 and right one from the point (0, 1). This would be (1, -1).
Alternatively you can get the point to the left of (0, 1). You would up 2 units and left one unit. This would yield the point (-1, 3).
Two points is enough the connect the dots. Three or four is okay too.
Example Two
Graph the line y = x/2 - 3.
From the above line, the y-intercept is the point (0, -3). The slope is one half or one over two.
With a slope of a half, the next point can be found by doing 1 unit up and then 2 units right. This gets the point of (2, -2).
Other Examples
Taxi Ride Example
One taxi company charges 5 dollars plus 1 dollar for every kilometer travelled.
A graph of this would have a y-intercept of 5 and a slope of 1. The equation of this line would be F = d + 5. The variable is the fare amount to be paid and d is the distance (number of kilometers).
If a rider has travelled 12 kilometers, then this rider would pay 5 + 12. This is 17 dollars.
Subscription Plan
Charlie signed up for a fitness gym plan. There is a signup fee of 20 dollars and he pays 50 dollars a month at this gym. How much money does Charlie end paying to this gym after 7 months?
The linear equation here is P = 50x + 20. The variable x is the number of months spent at the gym. As there a fee of 20 dollars, the y-intercept is 20.
To find the amount paid after 7 months, substitute x = 7. The amount paid would be 370 (50 * 7 + 20).
Closing Notes
Graphing linear equations combines algebra and plotting co-ordinates. The concept of this is not too difficult. The algebra parts could be difficult if the linear equation has fractions and negative signs.
Students who are not great at algebra or not great with plotting co-ordinates really need to review those topics. These are fundamental topics for early high school mathematics.
