1. AP Calculus AB/BC
1.1 Lines

2. Lines and equations of lines are used in numerous business and economic applications.
Linear functions are taught beginning with Algebra 1 and continuing all throughout Calculus.
Our first definition in Calculus is “Increments”. An increment can be thought of something being added, or changed.
In mathematics, an increment is the change in the x and y values from one point to another.

3. Let’s take two arbitrary points on the coordinate plane as shown.
= the change in x
= run y
= the change in y
= rise
So, the increments in the coordinates are:
And, the slope of a line is defined as: x O
If two lines are parallel, then they have the same slope (the slopes have the same values).
If two lines are perpendicular, then the slopes are opposite (negative) reciprocals, and their product is negative one.

4. Example 1
These increments can be used to find the slope of the line through points A and B.

5. ▬▬ Equations of Lines ▬▬
Slope-Intercept:
y = mx + b
m = slope
b = y-intercept
Point-Slope:
m = slope
If (a, b) is a point in the coordinate plane, then
x = a is a vertical line through (a, b), and
y = b is a horizontal line through (a, b).

6. Find the slope and y-intercept and graph the line: 3x + 4y = 12
Example 5
Find the slope and y-intercept and graph the line: 3x + 4y = 12
First, solve the equation for y.
This gives a y-intercept of 3 and a slope of -3 +4

7. Now, it’s your turn. Find the slope and y-intercept, then graph the line

8. Example
The pressure p experienced by a diver underwater is related to the diver’s depth d by the equation of the form p = kd + 1 (k a constant). When d = 0 meters, the pressure is 1 atmosphere. The pressure at 100 meters is atmospheres. Find the pressure at 50 meters.
Here’s what we know: p