2. Graphs of Linear Functions
It is very important for many math topics to know how to quickly sketch straight lines. When we use math to model real-world problems, it is worthwhile to have a sense of how straight lines "work" and what they look like.
We met this topic before in The Straight Line. The following section serves as a reminder for you.
a. Slope-Intercept Form of a Straight Line: `y = mx + c`
If the slope (also known as gradient) of a line is m, and the y-intercept is c, then the equation of the line is written:
`y = mx + c`
The line `y = 2x + 6` has slope `m = 6/3 = 2` and `y`-intercept `c = 6`.
Graph of the linear equation `y = 2x+6`.
b. Intercept Form of a Straight Line: `ax + by = c`
Often a straight line is written in the form ax + by = c. One way we can sketch this is by finding the x- and y-intercepts and then joining those intercepts.
Sketch the line 3x + 2y = 6.
The x-intercept (that is, when `y = 0`) is:
3x = 6
x = 2.
The y-intercept (that is, when `x = 0`) is:
2y = 6
y = 3.
Joining the intercepts `(2, 0)` and `(0, 3)` gives the graph of the straight line 3x + 2y = 6:
Graph of the linear equation `3x + 2y = 6`.
Slope of a Line
The slope (or gradient) of a straight line is given by:
`m=text(vertical rise)/text(horizontal run)`
We can also write the slope of the straight line passing through the points (x1, y1) and (x2, y2) as:
Using this expression for slope, we can derive the following.
c. Point-slope Form of a Straight Line: `y − y_1= m(x − x_1)`
If a line passes through the point (x1, y1) and has slope m, then the equation of the line is given by:
`y − y_1= m(x − x_1)`
Find the equation of the line with slope `−3`, and which passes through `(2, −4)`.
`m = −3`,
`x_1= 2`, and
So using the formula `y − y_1= m(x − x_1)`, the equation is
y − (−4) = −3(x − 2)
y + 4 = −3x + 6
y = −3x + 2
The graph is:
Graph of the linear equation `y = -3x+2`.