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Graphing Functions

A good approach to graphing a function is to make a table of a handful of possible inputs and outputs. We'll graph the points we get from our table and connect them according to the pattern we see.

When we make our table, the inputs ($x$) will be the $x$-values in our coordinate pairs, and the outputs ($f(x)$) will be the $y$-values.

Let's try an example. $$f(x) = -2x + 1$$ First, we're going to set up our table. It's a good idea to pick symmetrical values to plug in for $x$. This means, for example, if I decide to try out $x=2$, I should also include $x=-2$ in my table.

It's also a wise choice to include $x=0$ in every table. This gives you the $y$-intercept for your graph—where the line crosses the $y$-axis.

If you need help figuring out how to get output values from functions, refresh yourself!

Now, let's plot the points we've got on a graph. These points will be coming from the third column in our table.

Now, all we need to do is connect the dots in the pattern that we see.

In general, the graph of a function contains all the points that satisfy the equation of the function.

Ready to see an example?

Great! Remember, for ordered pairs, the $x$-term comes first, and that is what we input to the function. The $f(x)$ (or $y$) term comes second, and that is our output for the given input.

Here's the graph and the some points that satisfy the function.

There are infinitely many points and all of them make up this graph.

Another example is$$f(x)=x^2-x$$

Yet another example $$f(x)=\frac{1}{x}$$

Also notice that if the $x$ value is determined, there can be at most one $y$ value that satisfies the function. (That is actually how functions are defined! If you take in a value $x$, the function $f(x)$ gives only a value if $x$ is in its domain.) This will come in handy later.

Function Graphs

Here is a list of some basic functions and their corresponding graphs. Once you are familiar with graphing functions, you can start to solve systems of equations (like linear or linear-quadratic systems) using graphing!

Linear

$f(x)=x$

Quadratic

$f(x)=x^2$

Cubic

$f(x)=x^3$

Square Root

$f(x)=\sqrt{x}$

You can learn more about square root functions here.

Cube Root

$f(x)=\sqrt[3]{x}$

Absolute Value

$f(x)=|x|$

You can learn more about absolute value expressions here.

Reciprocal

$f(x)=\frac{1}{x}$

Natural Logarithmic

$f(x)=\ln x$

Exponential

$f(x)=e^x$

Cosine

$f(x)=\cos x$

Sine

$f(x)=\sin x$

Tangent

$f(x)=\tan x$

It's possible you haven't seen some of these graphs before - that's okay! This is a basic overview of many of the functions you'll see in the future as you continue on your math journey!

(Video) Introduction to Graphing Quadratic Equations

by Garret Cahill

In this video, Garret discusses how to graph quadratic functions. Before looking at this video, take a look at how to graph linear functions.

$\large{\text{Summary}}$

The equation of a quadratic is $$y=a(x-h)^2+k$$ When we graph the function $f(x)=x^2$, we get the basic parabola with the vertex at the origin.

When you have $f(x)=(x-h)^2$ the graph moves right $h$ units.

When you graph $f(x)=a(x-h)^2$ the values of the graph increase, with the vertex remaining the same.

When you graph $f(x)=a(x-h)^2+k$ the graph moves $k$ units. The vertex is always $(h,k)$.

The video then talks about $x$-intercepts which are where the graph crosses the $x$-axis. The line of symmetry is the line which the parabola is symmetric over.

When you are given a quadratic equation to graph in standard form, you can use the intercepts to plot the graph. You just have to plug in $0$ for $y$ to find the $x$-intercepts and plug in $0$ for $x$ to find the $y$-intercept.