Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.1 Functions and Graphing
Linear Functions are extremely basic functions but are also incredibly helpful because we can use them as a starting point when trying to identify patterns and test theories. In Calculus, we will often start with a linear function to practice our latest idea before moving on to curved lines.
Forms
Linear Functions are the most basic functions that we can investigate. A linear function is a function with one y and one x. There are three main forms for a linear equation:
Standard form finds most of its use with a topic called Linear Algebra which will be discussed later in these notes. Writing a linear equation this way is great for finding points of intersection between multiple equations, also called a System of Linear Equations.
Slope-Intercept form is named so because the equation easily identifies the slope, m, and the y-intercept, b. This form is easiest for graphing the equation because it gives you a starting point (0,b) and a rate of change, slope m, to work with.
Point-Slope could be the most useful for Calculus. That is because the point slope form can be used to find the equations of lines that are tangent to a curve. It is called Point-Slope because given a point (a,b) and a slope m, we can plug in right away in order to solve. Generally, one might plug into the point-slope form, solve, and rewrite in the slope-intercept form for graphing.
Slope
In order to calculate the slope of a linear equation we have a couple of strategies. One strategy involves writing the equation in slope-intercept form, then the slope is the number attached to the x. The slope in y=2x-3 is 2. Another strategy, and one that is used when we don’t know the equation for the line, involves finding two points and calculating the 
. The rise is the change in y-values and the run is the change in x-values. There are many ways to write this, many of which will be used throughout these notes.
The slope, in a way, tells us the direction that the line is going on a graph. If m<0 then the graph has a negative slope and is pointing downwards (to the right). If m>0 then the graph has a positive slope and is pointing upwards (to the right). If m=0 then graph is not going up or down, it is a horizontal line because there is no change increase or decrease. If m= 
then the slope is undefined and the graph is vertical. This case obviously doesn’t occur in linear functions because it would not pass the vertical line test, but it is helpful for defining vertical boundaries or what a curved line might do. A horizontal line would be written as y=b since the slope is 0, while a horizontal line would be written as x=a. The magnitude of the slope, |m|, defines how steep the line is. If |m| is close to 0 then the line is relatively flat, while a larger value for |m| indicates that the line has more change vertically.
Example 1: Write the following equation in Slope-Intercept form: y-3=4(x-2)
Solution: Our goal is to format this as y=mx+b, so we should isolate the y using the order of operations backwards.
Example 2: Find the equation of a line through (2, 5) with slope m =-7/2.
Solution: Using the point-slope form of a line y-b=m(x-a)
Example 3: Find the equation of a line through (3, 7) and (-2,-3)
Solution: First we find the slope using the change in y divided by the change in x, then use the point-slope form of a line
Graphing
The graph of a linear function is a straight line. The easiest way to graph a linear equation is to write it in Slope-Intercept form and use the y-intercept as a starting point. Recall that slope can be thought of as “rise over run” and use the slope to find the next point. Then we can connect the two points and extend in both directions.
Recall that the y-intercept occurs when x=0 and the x-intercept occurs when y=0.
Example 4: Graph y=3x-2
Solution:
Example 5: Graph y= -2/5x+4
Solution:
Example 6: Find the x and y intercepts for y=2x-11
Solution:
The x-intercept occurs when y=0, so 0=2x-11 and x=11/2
The y-intercept occurs when x=0, so y=2(0)-11 and y=-11
Domain
The domain of a linear function is always unrestricted. That is to say, there are no domain restrictions. This can be written as D: x∈
(Set Notation) or D: x∈(-∞,∞) (Interval Notation).
Range
The range of a linear function is also always unrestricted. Since y varies proportionally with x, we can see that any output is possible: R: y∈ or R: y∈(-∞,∞).
Special Forms
There are two special linear functions worth discussing:
Horizontal lines are linear functions of the form y=b. A horizontal line has a slope equal to 0. These are helpful to describe bounds, like the highest or lowest values.
Vertical lines are linear functions of the form x=a. A vertical line has an infinite slope. These are often used to represent vertical asymptotes.
Example 7: Graph y=7
Solution:
Example 8: Graph a vertical line whose x-value is 2.
Solution:
Example 9: What are the equations of the x-axis and y-axis?
Solution: The y-axis occurs as a vertical line at 0, so it is x=0, similarly the x-axis is y=0.
Comparing Equations
When comparing linear equations, there are three possible solutions:
1. There are zero points of intersection because they will never intersect. These linear functions are called parallel because they have the same slope.
2. There is one point of intersection because they have different slopes. This is the most common type of solution. There is a special case: lines are perpendicular to each other if their slopes are the negative reciprocal of each other. That is to say, if m1 and m2 are the two slopes then m1= (-1)/m2 . A perpendicular intersection means that the lines intersect at a right angle. This is incredibly useful for physics!
3. There are infinitely many points of intersection because they are the same line. If two equations are representative of the same line, then they intersect everywhere and all points are solutions the system.
Example 10: How many points of intersection are there between y=3x+4 and 2y-4x=6
Solution: It is helpful to write the equations in the same form. In this case, slope-intercept form is ideal:
We can see that the slope of the first equation is m=3 and the slope of the second equation is m=2. Therefore, the two equations must intersect at one point. This point can be found by setting the equations equal to each other.
Take this x value and plug it into either equation to find the y-value where they intersect.
So, the two equations intersect one time at (-1,-2).
Example 11: How many points of intersection are there between 4x-2y=6 and y=2x-6
Solution: If write the first equation in slope intercept form we get the following
We can see that the two equations have the same slope, but different y-intercepts. Thus, they are parallel and never intersect.
Review
A linear function is a function where y and x are directly related/proportional to each other. When graphed, they appear as straight lines.
Domain: D: x∈
Range: R: y∈
x-intercept: Occurs when y=0
y-intercept: Occurs when x=0
Lines that are parallel never intersect.
Lines that are perpendicular intersect at a right angle.
Equations
Standard Form: ax+by=c
Slope-Intercept Form: y=mx+b
Point-Slope Form: y-b=m(x-a)
A slope perpendicular to m is given by (-1)/m