How to Graph a Function

Every graphing calculator, whether it is a TI-84 Plus™, Desmos or the tool on this site, follows the same three-move routine: enter the equation, set the window, read the result. What changes between them is which key you press — not what you are actually doing.

Step 1 — Enter the equation

Open the function entry screen. On a handheld that is the Y= key; here it is the expression box in the panel. Type your equation in the form y = f(x).

A few conventions that trip people up:

• The ^ key raises to a power, so x^2 is x squared.
• Multiplication can be implicit — 2x works as well as 2*x.
• Close every bracket you open. An unclosed bracket is the single most common cause of a confusing result.

Step 2 — Set the viewing window

The window is the range of x and y values the calculator draws. Before you trust anything you see, check the window — a wrong window is behind almost every "my graph looks broken" moment.

The standard window runs from −10 to 10 on both axes. That is a sensible default for x^2, but a poor one for 100x, which shoots off the top almost immediately, or for 0.01x, which looks like a flat line.

If the graph is empty or a flat line, reset to standard first and only then decide whether you need to zoom in. Guessing at the window usually wastes more time than it saves.

Step 3 — Read the graph

With the curve on screen, drag the pointer along it to read exact coordinates. Switch on tracing to step through values one at a time — the fastest way to answer a question that asks for a specific input, such as "what is y when x = 3".

For a full column of values, open the table of values instead. It generates the same numbers in a grid, which is what most homework questions about functions are really asking for.

Reading shapes quickly

Recognising a shape tells you which transformation you are looking at before you read a single number:

• Straight line — linear. The steepness is the slope.
• U-shape or upside-down U — quadratic. The bottom or top point is the vertex.
• V-shape — absolute value.
• S-curve — cubic, or an exponential if it flattens at one end.
• Two separate branches — a rational function with an asymptote between them.

The three mistakes that make graphs look wrong

1. Zoomed in too far

A tiny slice of a curve is nearly straight, so x^2 at 0.001 scale looks like a flat line. Reset to standard before concluding anything.

2. Multiplying wrong

1/2x is not the same as (1/2)x. Without brackets it parses as 1/(2x), which is a completely different curve — one that shoots to infinity at zero. If in doubt, bracket it.

3. Forgetting to switch the function on

Most calculators let you toggle a row on and off. If a function is off it simply does not draw, and it is easy to assume the graph is broken when the row is just disabled.

Where to go next

Once you can plot a single function confidently, the natural next steps are graphing a parabola and piecewise functions, which add domain restrictions and endpoints.

If you are getting an error message instead of a graph, the TI-84 Plus™ error code guide covers every error the handheld can throw.