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Now that you possess a strong working knowledge of functions, you can start exploring various subsets of functions. For instance, linear functions are any functions that can be written in the format $f\left(x\right)=mx+b$ , where $m$ and $b$ are both real numbers and $m\ne 0$ . Linear functions always represent straight lines on the coordinate plane. One example would be $f\left(x\right)=2x-1$ , which fits the $f\left(x\right)=mx+\mathrm{b}$ format because you're adding $-1$ to $2x$ . In the special case where $m$ does equal zero this is called a constant function, and in some cases is considered linear so be sure to make sure you are using your class's definition of this term.

Linear functions generally have simple equations, but they will prove extremely important as you continue to study mathematics. Graphing linear functions is among the first things you will learn how to do.

The graph of a linear function will always be a straight line with either a positive or negative slope, depending on the value of m. If m is positive, you get an increasing line. If it's negative, you get a decreasing line. If m was zero, you would simply be looking at a horizontal line and not a linear function because the output does not scale with the output. Here is an example of what a graphed linear function looks like:

There are two ways to graph a linear function. The first is to substitute two random values for x and work out the equation to get two points. For example, let's say we're working with $f\left(x\right)=3x+1$ . We can sub 1 and 2 in for $x$ to calculate two points:

$f\left(x\right)=3\left(1\right)+1=4$

$f\left(x\right)=3\left(2\right)+1=7$

Thus, $\left(1,4\right)$ and $\left(2,7\right)$ must both be points on our line. From there, simply plot both points on the coordinate plane and connect them with a straight line extended in both directions. You're done!

Alternatively, you can graph a linear function using its slope (m) and y-intercept (b). Begin by plotting the y-intercept which is always $\left(0,b\right)$ . Returning to the example above, the y-intercept is $\left(0,1\right)$ .

Next, use m (slope) to calculate the rise and run by expressing it as a fraction. Since m is 3, we can express it as $\frac{3}{1}$ . The numerator is the rise, so the next point goes up three units vertically on the y-axis. The denominator is the run, so we go right by one unit on the x-axis. That makes our next point $\left(1,4\right)$ . If the rise was negative, you would count down instead of up. Likewise, you would count left instead of right if the run was negative. We can double-check our work by making sure the new point satisfies the equation:

$f\left(x\right)=3\left(1\right)+1=4$

Looks good! We now have two coordinates in $\left(0,1\right)$ and $\left(1,4\right)$ , so connect them with a straight line and extend it out in both directions. Unless otherwise noted, the domain and range of any linear function is the entire set of real numbers because there is a value x for every y and vice-versa. You can show this in your graphs by drawing a little arrow as you extend your lines.

a. Name two points on the line represented by $f\left(x\right)=2x-2$

$\left(-2,-6\right),\left(-1,-4\right),\left(1,0\right),\left(2,4\right)$ etc.

b. Name two points on the line represented by $f\left(x\right)=-3x+4$

$\left(-2,10\right),\left(-1,7\right),\left(1,1\right),\left(2,2\right)$ etc.

c. Name two points on the graph of a linear function where $m=2$ and $b=3$

$\left(0,3\right)$ and $\left(1,5\right)$

Solving One-Step Linear Inequalities

College Algebra Diagnostic Tests

Linear functions are a gateway to more advanced topics in mathematics, so your student cannot simply ignore the problem if they are struggling to keep up with their class. If you're concerned that the student in your life may be falling behind or simply wish to empower them to reach for their full potential, a 1-on-1 tutor can provide the personalized attention that isn't always available during school hours. Contact Varsity Tutors today to speak with a knowledgeable Educational Director and learn more about the potential benefits of tutoring.

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