Business Calculus: Slope definition and slope-intercept form of a straight line
Slope: Slope of a straight line is its inclination relative to the horizontal axis. Slope
of line that goes through two distinct points (x1 , y1 ) and (x2 , y2 ) is defined as
m=
y2 − y1
x2 − x1
Slope has many different explanations; here are some:
(a) Rise over run
(b) Change in y over change in x sometimes written as
∆y
∆x
(c) Rate of change of y with respect to x
(d) Derivative of a function or instantaneous rate of change
(e) Tangent line identified by the ratio of change in y and change in x
(f) Relative changes between two variables; for example how one’s income changes as that
person’s education changes
While all of the above are different explanations of slope or rate of change, it is very
important to think of slope as rate of change which gives more real world perspective of
slopes.
The following problems ask you to find the slopes and interpret the results:
Problem 1: Find the slope of the straight line that goes through the points (2, −1) and (−5, 7) and
interpret the slope.
3
3
Problem 2: Find the slope of the straight line that goes through the points (− , 3) and (4, − )
2
4
and interpret the slope.
Problem 3: Write the equation of horizontal line and find its slope and explain.
Problem 4: Write the equation of a vertical line and find its slope and explain.
1
The slope-intercept form of a straight line
The slope-intercept form of a straight line is the most useful form of the equation of a straight
line. Many real world problems may be modeled using the slope-intercept form of a straigt
line. It has various real world and scientific applications in science, social science, business,
and many other fields.
The slope-intercept form of a straight line is y = mx + b where m is the
slope and b is the y-intercept.
10
8
6
4
y = −2x + 4
2
−3
−2
−1
1
2
3
−2
Figure: The slope-intercept form of the straight line y = −2x + 4
Here the y-intercept of the line is at (0, 4) and the slope of the line is −2 which means that
as you move from left to right on the graph the line goes down. Similarly, for a line with
positive slope, as you move from left to right the line goes up. Also remember the special
cases, a horizontal line has slope zero, a vertical line has undefined slope, and a line through
the origin has y-intercept b = 0, that is the line goes through the point (0, 0) and has the
general form y = kx where k is the slope of the line.
Problem 5: Express the equation 2x + 3y = 6 in the slope-intercept form to find the slope and
coodinates of the y-intercept. Interpret the slope in this context.
Problem 6: A home was purchased for $300,000 this year. Every year the price of the home
increases by 3% of this purchase price. Create a linear model for the price of the home
as a function of time in years.
Problem 7: A car was purchased this year for $25,000. Its price depreciates by %5 of the purchase
price every year. Create a linear model for the price of the car as function of time in
years.
Problem 8: Your monthly cell phone bill is $50 that includes 2 Gigabytes of data, but any additional
data is charged at $10 per Gigabytes. Create a linear model for this scenario.
2