BackSlopes and Equations of Lines in Business Calculus
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Part I: Slopes
Definition of Slope
The slope of a nonvertical line is a measure of its steepness, defined as the ratio of the change in y to the change in x between two points on the line. The slope is denoted by m and calculated as:
Formula:
Where and are two distinct points on the line.
The slope of a vertical line is undefined.
Note: If , then (the slope).
Examples of Calculating Slope
Example 1: Find the slope of the line passing through and .
Example 2: Find the slope of the line passing through and .
Example 3: Find the slope of the line passing through and .
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Interpretation of Slope
A line with a positive slope rises from left to right.
A line with a negative slope falls from left to right.
A line with zero slope is horizontal.
A line with undefined slope is vertical.
Equations of Lines
Vertical and Horizontal Lines
The vertical line passing through has the equation , where is the x-intercept.
The horizontal line passing through has the equation , where is the y-intercept.
Point-Slope Form
The point-slope form of the equation of a line with slope passing through is:
Example: Find the equation of the line passing through with slope .
Example: Find the equation of the line passing through and .
Slope-Intercept Form
The slope-intercept form of a line with slope and y-intercept is:
Example: Find the slope and y-intercept of the line .
Slope: , y-intercept:
Example: Find the equation of the line containing and in slope-intercept form.
General Form
An equation of a line is in general form if it is written as:
Where , , and are real numbers, and and are not both zero.
Example: The equation is in general form with , , .
Intercept Form
The intercept form of a line with x-intercept and y-intercept (where ) is:
Example: Find the equation of the line with x-intercept 3 and y-intercept -4.
Parallel and Perpendicular Lines
Two distinct nonvertical lines are parallel if and only if their slopes are equal: .
Two lines (neither vertical) are perpendicular if and only if the product of their slopes is :
The slopes of perpendicular lines are negative reciprocals.
Special Cases:
Any two vertical lines are parallel.
A vertical line is perpendicular to a horizontal line.
Example: Find the equation of the line passing through which is parallel to (vertical line): (vertical line through ).
Perpendicular to : (horizontal line through ).
Example: Find the equation of the line passing through which is parallel to .
Parallel slope:
Perpendicular slope:
Application: Linear Modeling with Real Data
Linear equations are used to model real-world data, such as the cost of tuition over time.
Example (Tuition): The table below lists the annual cost (in 2019 dollars) of tuition and fees at private four-year colleges for selected years.
Year (2000+t) | Tuition and Fees (T) | Year (2000+t) | Tuition and Fees (T) |
|---|---|---|---|
2010 | 31,500 | 2016 | 35,720 |
2012 | 32,470 | 2017 | 36,380 |
2014 | 33,690 | 2018 | 36,330 |
2015 | 34,570 | 2019 | 36,880 |
Let correspond to the year 2000.
Use the points and to determine the linear equation that models the data.
Finding the Linear Model:
This equation models the increase in tuition and fees over time.
Graphing: The data and the linear model can be plotted to visualize the trend.
Additional info: The notes include MATLAB code for plotting and fitting the data, which is a common application in business calculus for modeling and prediction.
