How to Find Intercepts on a TI-84

An x-intercept is the point where a curve crosses the horizontal axis; a y-intercept is where it crosses the vertical axis. On a TI-84 these are two separate jobs. The x-intercepts, which your teacher may call zeros or roots, are found by a search routine that hunts for a sign change. The y-intercept is found by evaluating the curve at one particular value of X.

Students often lose time because they expect one menu item to hand over both. It will not, and the reason is mathematical rather than a quirk of the interface: a root is a value of X that has to be searched for, while the y-intercept is simply the value of the function at X = 0.

This page uses Y1 = X² − 4 as the working example, because it has two roots and an obvious negative intercept on the vertical axis, so each readout is easy to check by eye.

The CALC menu behind 2nd ▸ TRACE

Both routines sit in the same menu, which opens when you press 2nd and then TRACE.

The menu only works once a function has been drawn, so press GRAPH first. If the menu opens but the searches behave strangely, the usual cause is that the current window hides the crossing you are looking for.

Finding the x-intercepts with 2:zero

Press 2nd TRACE, choose 2:zero, and the calculator asks for three pieces of information in turn: a left bound, a right bound and a guess.

1. Move the cursor with the arrow keys so it sits to the left of the crossing, then press ENTER to set the left bound.
2. Move the cursor to the right, past the crossing, and press ENTER to set the right bound.
3. Move the cursor close to the axis and press ENTER to confirm the guess.
4. Read the answer: the screen reports the root as a value of X with Y equal to zero.

For the left-hand root of Y1 = X² − 4 the readout is X = −2, Y = 0. The right-hand root gives X = 2, Y = 0, and you have to run the routine a second time to reach it, because each pass finds one crossing only.

The two bounds must sit on opposite sides of the axis. The routine is looking for the point where the curve changes sign, and if both bounds are above the axis, or both below it, there is no sign change between them to find.

The guess matters less than the bounds, but it does matter. It tells the calculator where to start looking inside the interval you have defined, so placing it near the crossing is quicker and more reliable than leaving it wherever the cursor happened to be.

Finding the y-intercept with 1:value

The vertical intercept is a single evaluation rather than a search. Press 2nd TRACE, choose 1:value, and the calculator prompts for a value of X. Type 0 and press ENTER.

The bottom of the screen shows X and Y, so the coordinate of the intercept can be read straight off without any algebra. The same option is useful well beyond intercepts: typing any value of X returns the corresponding Y, which makes 1:value a quick way to test a point against the curve.

Choosing the window before you search

The window decides whether the routine can find anything at all, so it is worth setting deliberately rather than leaving it wherever a previous session ended.

• ZStandard (ZOOM 6:ZStandard) resets the view to −10 to 10 on both axes with a scale of one unit. It is the fastest way to see a whole polynomial at once, and the intercepts of most textbook quadratics and cubics fall inside it.
• ZDecimal (ZOOM 4:ZDecimal) sets X from −4.7 to 4.7 and Y from −3.1 to 3.1 on the TI-84 Plus, so the screen steps exactly one unit of X at a time. On a colour-screen CE the limits are a little wider to keep the same pixel spacing. This window is the one to use when you want to read coordinates directly from the grid.

If a crossing sits close to the edge of the screen, 6:ZStandard puts it back in view. If two roots are so close together that they share a pixel, 2:Zoom In around the axis separates them.

When the calculator refuses: ERR: BOUND

If you set both bounds on the same side of the axis, or place the guess outside the interval you have defined, the routine has nothing valid to search and reports ERR: BOUND. Press 1:Quit, run 2:zero again and widen the interval. One bound should clearly be above the axis and the other clearly below it.

Graphs that have no x-intercept

Not every curve crosses the horizontal axis, and the search routine cannot invent a root that does not exist. Y1 = X² + 1 sits entirely above the axis, so no bracket will ever produce a root. Similarly, a curve such as Y1 = √(X) stops at the origin and has no left-hand branch, because the square root of a negative number is undefined — the same restriction that produces ERR: DOMAIN when you evaluate it on the home screen.

Reading the shape of the graph first tells you which case you are in. A curve that never reaches the axis has no root to find, and recognising that is faster than setting bounds that cannot work.

Cross-checking your intercepts

Two quick checks confirm an intercept without any extra maths. The TABLE screen, opened with 2nd GRAPH, shows a column of X values and the matching Y1 values, so the sign change either side of a root is visible at a glance; our guide to the TABLE feature covers the setup. And if you want to confirm the algebra behind a curve before hunting for its crossings, graphing the function first is the sensible order.

On test day, bear in mind that the ACT allows a permitted calculator such as the TI-84 on the mathematics test only, and the SAT allows a graphing calculator on all of its maths questions while prohibiting models with a computer algebra system. The rules are revised occasionally, so check the current wording with the board rather than relying on memory. Our ACT calculator policy page sets out what applies.