linear functions

Linear Functions In Algebra

A function is often used to model data that is important to us. It is useful in the sense that it can allow us to somewhat predict what will happen in the future mathematically.

Introduction

A function or mathematical model will not ever be exact. So a linear function is really just an approximation. With that said, linear functions and models of data can be very useful. You see them often in graphs. They are the straight lines you see representing data. A linear function usually takes the form \(f(x) = ax + b \). A line from this function on a graph can intersect one or both of the axes. Lines like this can have a slope. Slope is shown as :

$$ \frac{\Delta y}{\Delta x} $$.

If the line crosses the y-axis it is called the intercept. If we have a linear function \(f(x) = 2x + 5 \), 2 is the slope and 5 is the y-intercept.

The x-intercept is where the line crosses the x-axis. If we fill out the equation with intercepts and slope it should equal out to zero. This is how you can check your answer. The x-intercept is also called the <zero> of the function. If the slope of a function is not zero, then the graph only has one x-intercept.

Modeling Functions

Functions are often used to model data. When we have a graph of data, the slope also tells us its rate of change. This is useful if it is changing at a constant rate. The constant rate of change is the slope of a straight line on a graph. So basically, if the slope is the same between two points then you can use a linear function. An exact model will be able to describe the data very closely, if not exactly. We usually never see exact models of anything outside of a textbook so keep that in mind.

Linear Regression

The example I have mentioned have all been regression problems. This is because we have used one variable to predict the value of another variable. This is the very definition of linear regression.

Equation Of A Line

As discussed before, any quantity that increases at a constant rate, can be modeled with a straight line. The slope of this line can be computed easily. The \( \Delta y = y - y_1 \) and \(\Delta x = x - x_1 \) are what is needed. In fact:

\[ \frac{ \Delta y}{ \Delta x} \] will give us the slope itself. This is the slope formula. It also has other versions.

They look like this:

\[ m = \frac{ y - y_1}{x - x_1} \]

and

\[ y = m(x - x_1) + y_1 \]

Point-Slope Form

You will use the last two equations more than the first. The first is just a generalization. The latter two are just different forms but say the same thing. The last one is also known as the point-slope equation. So here is an example I found to show how this works.

Example

Find an equation of the line passing through the points (-2, -3) and (1,3).

You will start by finding the slope or rate of change. If you are looking at a graph, this will be the line you are looking at.

\[ m = \frac{3 - (-3)}{1 - (-2)} = \frac{6}{3} = 2 \]

That gives us a slope of 2 for this line. The line consists of the two points mentioned above. Now we just use one of the points along with our slope and put it in the point-slope form. I will use the first point given.

\[ y = 2(x + 2) - 3 \]

Slope-Intercept Form

The slope-intercept form is another variation. It is \( y = mx + b \). M is the slope and b is the y-intercept.

Example

Find a point-slope form for the line.

First we have a slope of \(-\frac{1}{2}\) passing through the point (-3, -7).

The point-slope form is \( y = m(x - x_1) + y_1 \). Just plug in the values that we were given. We get:

\[ y = -\frac{1}{2}(x +3) -7 \].

We get the slope-intercept form by just simplifying this expression.

\[ y = -\frac{1}{2}x - \frac{3}{2} -7 \]

\[ y = -\frac{1}{2}x - \frac{17}{2} \]

Intercepts

An equation of a line has another form which is called the standard form. It looks like: \( ax + by = c \). This is the third equation of a line we have talked about now. So we have in total: point-slope form, slope-intercept form, and now the standard form. To find the intercepts from a standard form equation, you will set y or x to zero and solve the equation. If y is set to zero then you will get the x intercept. Setting x to zero will then give you the y intercept.

Example

Find the intercepts for the equation \( 4x + 3y =6 \).

Start with setting y=0.

\( 4x + 0 = 6 \)

\( x = 6/4 \)

\( x = 3/2 \) = x-intercept

Now let us set x to zero to find the y intercept.

\( 0 + 3y = 6 \)

\( y = 6/3 \)

\( y = 2 \) = y-intercept

Other Types Of Lines

There are even more types of lines you can see on a graph. Don’t worry, they are all easy. The first is a horizontal line. On a graph, this represents a constant function. It is the easiest to recognize and do anything with. A horizontal line has the same y coordinate but different x coordinates. It also has a zero slope.

A vertical line is the opposite. It will have the same x coordinate but different y coordinates. Its slope will be undefined.

Parallel lines are the next type. They can’t be vertical to be considered parallel. These lines also need to have the exact same slope. If these conditions are met they will be considered parallel.

Perpendicular lines are the last type. For two lines to be perpendicular they need to be negative reciprocals.

Related Data

This may seem like common sense, but it is also important to state. Data can be related to each other. If one quantity changes then another will change because it can be directly related. However, it is important to be able to see if data changes because it is related or not.

Recapping It All

Point-slope form = \( y = m(x - x_1) + y_1 \). This is used to find the equation of a line if you have two points or one point and a slope to work with.

Slope-intercept form = \( y = mx + b \). This is a discrete equation for any certain line. It is obtained by using the slope and and the y-intercept.

Interpolation = Estimated values that are between two data points.

Extrapolation = Estimated values that are not between two data points.

In this section I covered more of the basics, which include line equations and working with data points. The next chapter will be over linear equations and working with them.