Now that we have investigated functions from an algebraic perspective, how can we also investigate functions from a geometric perspective? The answer is to use a graph, and the idea of connecting geometry with algebra in mathematics was one of the most significant developments in the history of mathematics. The credit for this groundbreaking idea generally goes to two different French mathematicians who lived roughly in the 1600s: Pierre de Fermat and René Descartes. Actually, both men have rather interesting stories: neither was a professional mathematician for their entire career (Fermat was a lawyer and Descartes was a soldier), which perhaps shows that important mathematical discoveries aren’t always made by people working only math jobs.

Imagine for a moment that the only way you could work with a function was by using algebra. This is pretty tough for us – if I had to guess, you probably had some graphs in mind while reading the previous section! How would you visualize a function? What kinds of visual intuition might you have about functions if all you knew about was algebra? This was actually the state of mathematics for a long time! Try to put yourself in the mind of Fermat and Descartes: what is the first thing you would need in order to draw a picture of a function?

If you said “a coordinate system”, then you’ve come up with Fermat and Descartes’ big contribution to this subject!

The rectangular coordinate system

When we think about functions, we start by thinking about the set of inputs, or the domain, for our function. Typically these inputs are numbers, and we’ve seen that often these numbers come from the real numbers. In fact, most of the time the inputs can come from any real number. When we draw the set of all real numbers, we most often represent this with a number line.

While we can draw number lines very generally using the information above, a very common way to draw a number line for a graph is to first mark zero, and then our one unit length is given by where we place the tick mark for the number $1$.

But, inputs aren’t the only thing we have with functions: we also have outputs. Our outputs are also usually numbers, and so we could also represent them using another number line. But the outputs aren’t completely separate from the inputs, they depend on what input we have. We want to represent this relationship, so we are going to make this second number line vertical instead of horizontal. In other words, if we are given an input, we draw the corresponding output directly above that input, using the vertical number line.

You might be ready to complain about the drawing I just made, but before I draw one that looks a little more like what you were expecting, notice that what I did draw satisfies the requirements that I need in order to draw a function. I have a horizontal number line (called the horizontal axis) and I have a vertical number line (called the vertical axis) and they meet at right angles. I could use this set of number lines (called axes) to graph a function if I wanted to, by playing dots on the output values corresponding to my input values.

Of course, it is much more common to place the intersection of the two axes at zero on both number lines. When we do so, we typically call the point which is zero on both number lines the origin. The axes below are probably more what you are used to seeing, and the origin is located at the intersection of those axes.

While we are on the subject of vocabulary, we will sometimes use $x$ to represent the input values of our function, and so the horizontal axis is also called the $x$-axis. We will sometimes use $y$ to represent the output values of our function, and so the vertical axis is also called the $y$-axis. The variable $y$ represents the function’s output values, so we will sometimes write $y=f(x)$ to connect the $y$-values to the idea of the function. The $y$-variable is also sometimes called the dependent variable because its value depends on the input. The $x$-variable is then called the independent variable because its value doesn’t depend on anything; we get to choose the inputs. As much as we aren’t tied to placing the horizontal axis at $y=0$ we are also not tied to using $x$ and $y$ as our variables! If we used $t$ for the independent variable (the inputs), then the horizontal axis would be called a $t$-axis. Overall, the idea of using a coordinate system to do geometry is called analytic geometry. These coordinates are called rectangular coordinates because of the fact that the axes are at right angles to one another, and they are also called Cartesian coordinates in honor of Descartes. (Sadly, Fermat doesn’t get as much credit as Descartes!) This vocabulary lesson isn’t here because we want you to use all of these terms all of the time (though you are welcome to if they help you) but more because we want you to make sense of some vocabulary you might have heard in the past and connect it to the ideas we are discussing now.

These two number lines set at right angles create a plane which is called the coordinate plane.

We always assume that we are using rectangular coordinates unless we say otherwise. (Wait, there are other types of coordinates?? Yes! But we won’t talk about them here. Please feel free to look it up if you’re interested.) There is also a way to define a 3D coordinate space, but we would need three axes set at right angles to do so. In fact, you could define a 4D coordinate space, a 5D coordinate space, or larger, but these are hard to visualize. Here is a picture of the axes for 3D coordinate space. Here the origin is where the axes intersect, but it’s not labeled so that the markings on the tick marks are easier to see.

Now that we have a coordinate plane, we can describe any location in our plane using the $x$ and $y$ axes (and also the $z$-axis if we are working in 3D space). Say we have our axes, and we start at the origin and move $3$ units in the positive $x$-direction, and $1.5$ units in the positive $y$-direction.

Let’s look at this on our axes.

The solid red line shows us beginning at the origin and moving $3$ spaces along the positive $x$-axis, while the dashed blue line shows us starting at the end of the red arrow and moving vertically $1.5$ spaces. Notice that the red and blue lines are at right angles to one another, just like the $x$ and $y$ axes are. This means that we are “following” the $y$-axis when we move vertically, even if we aren’t actually touching the $y$-axis. (Remember that it doesn’t actually matter where we place the $y$-axis, so we could actually place it at $x=3$ if we wanted to!) We have labeled the location at the end of the blue arrow as $(3, 1.5)$. We use parenthesis to indicate that this is a point or location in the plane, and then the first entry (or coordinate) tells us how far we moved in the $x$-direction, while the second entry (or coordinate) tells us how far we moved in the $y$-direction.

We now have all of the pieces we need in order to draw a graph of a function.

It’s time to tackle an example! It’s incredibly important to remember that this graph, even though it looks like a line, is made up of all of the points $(x, f(x))$ that we get from evaluating our function. It can be easy to forget this when we start by drawing a line rather than by drawing a bunch of points!

We have motivated coordinate systems by wanting to draw graphs, but they are actually much more useful than that. Any time you want to be able to describe all of the points in a certain space, you can add coordinates. You’ll start with number lines, just like we have, so you’ll need a zero and a unit length to draw each number line. Then, set up your number lines at right angles (however many of them you need depending on the dimension of the space you’re working in) and you can use these ideas to describe all the points in your space.

Interpreting graphs

Now that we understand how to draw a graph, what kind of information can we get about functions from a graph? Let’s look at an example.

While we had some multiple choice prompts to help us analyze the graph in the previous example, in general we will expect you to be able to use a story situation to describe what you are seeing in the graph. You might want to go back to the previous example and write the correct multiple choice options in your own words to make sure that you are ready to do this on your own in future problems.

Let’s also pause and collect some of the vocabulary we used to describe parts of the graph above.

Notice that a graph can have many $x$-intercepts, but if the graph is a graph for a function it should have only one $y$-intercept since it should have only one output when $x=0$. In fact, if you have heard of the Vertical Line Test in a previous course, this test simply says that the graph of a function should have exactly one output for every input, so if you draw a vertical line from any output it should only pass through a single point on the graph. That single point on the graph represents a single output corresponding to that input. But don’t be afraid to analyze graphs that don’t represent functions! They are still often very interesting. Minimum and maximum values can give us quite a bit of information about a function. We can find them algebraically using techniques from calculus, or we can estimate them by looking at a graph of the function. In fact, one of the reasons that calculus is important is that it can help us use algebra to be precise about what we are seeing in a graph. When we talk about a function which is increasing, decreasing, or constant on some interval, we mean that it is increasing, decreasing, or constant on that entire interval. So, if you have an increasing function, if you pick any two inputs, the larger input has to have a larger output.

We can also talk about how the graph is increasing or decreasing: is it increasing slowly? Quickly? What might those things mean for the story problem associated with a graph? Is the function increasing more and more quickly? Increasing more and more slowly? These questions are associated with what we could call the concavity of a graph, in case you have seen that terminology before.

As we investigate functions using both their equations as well as their graphs, pay attention to things which are easier to know using an equation and things which are easier to know using a graph. It’s very powerful to have both tools at our disposal, and it can be incredibly useful to go back and forth from one representation to the other.

There is one more feature of graphs of functions that we should pay very careful attention to, and that is how the domain of a function is represented in a graph. Remember that the domain is the set of all input values for a graph, and we want to be sure that we are representing the domain accurately. Let’s use the next example to see how this works.

Graphs of linear functions

To wrap up this section, let’s look at the graphs of linear functions. Perhaps you remember that we defined a linear function as one whose equation could be placed in the form $y=mx+b$ for some values of $m$ and $b$. Let’s use a specific example to investigate what this means for its graph.

So, now we know that if we write an equation in the form of a linear function $y=mx+b$, the graph has to be a straight line. But the opposite is also true: if a graph is a straight line, the equation of that graph can be written in the form $y=mx+b$ for some $m$ and $b$.

Now we know that our two definitions for linear functions are really the same idea, so it makes sense to use either definition interchangeably. Linear functions and ratios and similar triangles are closely related ideas, so as you talk with kids about any of these ideas, you are helping lay the groundwork for deeper understanding of the other topics later in their mathematical journey.

2025-06-17 01:25:18