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Linear Functions

The name "linear functions" has the word "line" in it, which would suggest that, graphically, linear functions would look like lines.

And indeed they do. Here is a graph of: $$f(x)=-2x+3$$

Linear functions are straight lines because the function increases or decreases the same amount for every unit of change.

You'll see this a lot in science experiments, where there's a constant relationship between an independent variable and dependent variable.

Graphing Linear Functions

To graph linear functions, we need to plot at least $2$ points and connect them. We can choose any two points on the line so the goal is to pick easy to find ones.

Let's see how with an example. Suppose we want to graph: $$f(x)=3x-5$$ One of the easiest points to find is when $x=0$; we can simply throw out the $x$ term. $$\begin{align} f(x)&=3x-5\\ f({\color{#8925ae}{0}})&=3({\color{#8925ae}{0}})-5\\ f(0)&=-5 \end{align}$$ We now have the point $(0,-5)$.

Choosing $x=1$ for the second point is another easy calculation. $$\begin{align} f(x)&=3x-5\\ f({\color{#8925ae}{1}})&=3({\color{#8925ae}{1}})-5\\ f(1)&=3-5\\ f(1)&=-2 \end{align}$$ This gives us our second point of $(1,-2)$.

This is all we need to graph the function. Just draw the line that passes through these two points.

(Video) Basic Linear Functions

by Math Antics

Here's a great video by Math Antics that teaches you about the graphs of linear functions. He explains how both the slope and $y$-intercept affect how a graph looks.

$\large{\text{Summary}}$

How to Graph a Linear Equation

Linear equations are very easy to graph. All you need is $2$ points. The linear equation will always be the line that you make by connecting the points.

To find the points, we can use a table. Let's look at an example using the most basic linear equation: $y=x$. This equation says that whatever our input $x$ is, our output $y$ will be the same thing. We set up our input-output table like so: $$\begin{array}{c|c} x&y\\ \hline \\ \\ \\ \end{array}$$ Next, we write down some common input, $x$, values to plug in. You only need two to construct the line, but for now we'll use more. $$\begin{array}{c|c} x&y\\ \hline 1&\\ 2&\\ 3&\\ 5& \end{array}$$ Plug these $x$ values into our equation and solve to get their outputs or $y$ values. $$\begin{align} y&=x\\ \hline y&=1\\ y&=2\\ y&=3\\ y&=5 \end{align}$$ Because the input is equal to the output, we get the same numbers back out. Therefore, our table becomes: $$\begin{array}{c|c} x&y\\ \hline 1&1\\ 2&2\\ 3&3\\ 5&5 \end{array}$$ Let's plot these coordinates on a graph.

Finally, once you have your points, just draw a line through them to get the graph of the equation.

$y=mx$

Here, we have a similar equation to before, but now there an extra variable, $m$. For graphs, $m$ refers to the slope, or steepness. You'll learn how to calculate slope later, but for now let's see how it affects our graph.

When $m=1$, our equation becomes: $$y=1x$$ This is same as $y=x$ which is what we graphed before.

$m>1$

When $m=2$, our equation becomes: $$y=2x$$ If we fill out an input table, we get: $$\begin{array}{c|c} x&y\\ \hline 1&2\\ 2&4\\ 3&6\\ 4&8 \end{array}$$ For every $x$ value, the $y$ value gets doubled. Plotting this, we get:

This graph, is steeper than our previous one, which is in the background for reference.

As we pick larger and larger $m$ values, our graph gets steeper and steeper.

As $m$ gets larger and larger, the line becomes more and more vertical. But, we can never reach perfectly vertical.

$0<m<1$

To get less steep graphs, we pick an $m$ that's between $0$ and $1$. Let's let $m=\frac12=0.5$. Our input table gives us: $$\begin{array}{c|c} x&y\\ \hline 1&0.5\\ 2&1\\ 3&1.5\\ 4&2 \end{array}$$ And our graph gives:

As we choose $m$'s closer and closer to $0$, our slope get's less and less steep.

As $m$ gets closer to $0$ the line becomes more and more horizontal. When $m=0$, we do get a horizontal line.

$m<0$

We can have negative slopes as well! These will appear as mirror images of their positive counterparts.

Let's look at $m=-1$. This gives us the equation: $$y=-1x$$ Our input table is: $$\begin{array}{c|c} x&y\\ \hline 1&-1\\ 2&-2\\ 3&-3\\ 4&-4 \end{array}$$ And our graph is:

Similarly, a more negative slope makes a more steeper line, and a negative slope closer to $0$, makes a less steep line.

The Origin

No matter what we set $m$ to, our graphs will always have one point in common: the origin $(0,0)$.

$y=mx+b$

So what if we want a line that doesn't go through the origin? Well, we need to add a new variable, $b$. $$y=mx+b$$

To see what it does, let's start my letting $m=1$. Our equation will then be $y=x+b$.

If we let $b=1$, we have: $$y=x+1$$ Our input table is: $$\begin{array}{c|c} x&y\\ \hline 1&2\\ 2&3\\ 3&4\\ 4&5 \end{array}$$ The graph looks like:

As you can see, this line is parallel to $y=x$. If you look at where the point would cross the origin, $(0,0)$, it now crosses the point $(0,1)$. This $1$ is what we set $b$ equal to.

Plugging in other values for $b$, we get:

Solution

All the input tables have the inputs of $0$, $1$, $2$, and $3$. So let's plug these into our function and see what we get back out. $$\begin{align} f(x)&=2x+3\\ f({\color{#8925ae}{0}})&=2({\color{#8925ae}{0}})+3\\ f(0)&=3 \end{align}$$ I'll work through the others, but notice that already only the second answer choice matches this input-output. Therefore, it is the correct answer. $$\begin{align} f(x)&=2x+3\\ f({\color{#8925ae}{1}})&=2({\color{#8925ae}{1}})+3\\ f(1)&=2+3\\ f(1)&=5 \end{align}$$ $$\begin{align} f(x)&=2x+3\\ f({\color{#8925ae}{2}})&=2({\color{#8925ae}{2}})+3\\ f(2)&=4+3\\ f(2)&=7 \end{align}$$ $$\begin{align} f(x)&=2x+3\\ f({\color{#8925ae}{3}})&=2({\color{#8925ae}{3}})+3\\ f(3)&=6+3\\ f(3)&=9 \end{align}$$

Let's use the table to plot the coordinates and draw the graph of the function $f(x)=2x+3$.

Graphing Linear Functions

Linear functions are basically functions made of straight lines! We'll see this concept a lot of algebra, so it's important to get a solid understanding of linear functions.

Linear functions are different from non-linear functions, like quadratic or cubic functions. Linear functions have $x$ values. Quadratic functions use $x^2$, and cubic functions $x^3$. You'll learn about these two later!