How to Find a Tangent Line on a TI-84

You have a TI-84 in your hand for a reason: it is the calculator allowed in class and on the AP Calculus exam. Finding the tangent line comes down to three distinct tools on the machine. The one you choose depends on whether you need the line drawn, the slope only, or the equation for your free-response work. Each method is explained end-to-end, then you will see where the calculator's numeric answer can trick you.

Before you can draw a tangent line, the function must be visible on the screen. Press [Y=] and enter your equation. For the window, press [WINDOW] and set Xmin and Xmax so the point of tangency is roughly centred. A common failure: the tangent line is drawn, but the point is off the visible graph. The calculator does not warn you, it just draws nothing useful. Always verify that the x-value where you want the tangent sits inside the X-range and that the function's y-value at that x fits inside the Y-range. If you are working with a trigonometric function, check RADIAN vs. DEGREE mode. The TI-84 defaults to Radian; if your window is set to, say, 0 to 6.28, but the calculator is in Degree mode, the graph will look wrong and the tangent line will be meaningless. Press [MODE] and confirm the correct setting before you move on.

This is the fastest method when you need the line drawn on the graph and its equation displayed. From the graph screen, press [2nd] [DRAW] then 5:Tangent(. The calculator shows the graph and a prompt to enter an x-value. Type the x-coordinate of your point of tangency and press [ENTER]. The TI-84 draws the tangent line and displays its equation at the bottom of the screen in the form y = mx + b. The slope m is the derivative at that point, computed internally by nDeriv(. Write the equation down immediately, the display disappears as soon as you press any key. If the equation is cut off, press [TRACE] and use the left/right arrows to scroll. Source: TI-84 Plus CE Guidebook (Texas Instruments, 2019), DRAW Tangent( section.

When you only need the slope, for example, to write the equation in point-slope form by hand, use the CALC dy/dx tool. This method also works on the graph. Press [2nd] [CALC] then 6:dy/dx. The calculator prompts you to select a curve (if you have more than one) and then asks for an x-value. Type the value and press [ENTER]. The screen shows a blinking cursor on the curve at that x and prints the numeric derivative below. This is the same slope that DRAW > Tangent( uses, but without the line or equation. Use it when you want to keep the graph uncluttered or when you are checking a hand-calculated derivative. Source: TI-84 Plus CE Guidebook, CALC dy/dx section. Note that CALC dy/dx also uses a symmetric difference quotient, so the same caveats about differentiability apply.

If you are not on the graph screen, during a test or when building a program, use nDeriv( on the home screen. Press [MATH] then 8:nDeriv(. The syntax is nDeriv(expression, variable, value). For the derivative of f(x) = x² at x = 3, you enter nDeriv(X², X, 3) and press [ENTER]. The calculator returns 6. This command works anywhere on the TI-84: in the home screen, inside a program, or even as part of a larger expression. The default tolerance for the symmetric difference quotient is 1E-5, meaning the calculator uses h = 0.001 internally. Source: TI-84 Plus CE Guidebook, nDeriv( section. This is the same engine used by both DRAW > Tangent( and CALC dy/dx.

The TI-84 does not do symbolic algebra. When you ask for a tangent line, the calculator computes a decimal approximation of the slope, not an exact expression. For f(x) = sin(x) at x = π/4, the exact slope is √2/2, but the TI-84 displays a decimal. This is fine for checking your work or for numerical answers on a multiple-choice question, but it is not acceptable for free-response AP Calculus problems where exact values are required. Always round your calculator output to three or four decimal places unless the problem states otherwise. If the exact answer is a simple fraction, the decimal may still look correct: 0.5 for 1/2. But when the exact slope involves radicals or π, the decimal is an approximation. Write the exact form from your own derivative calculation, then use the calculator decimal to verify it.

Point Off Screen

If you enter an x-value and the tangent line does not appear, or the calculator beeps and returns to the graph without drawing anything, the point is outside the current window. Press [WINDOW] and adjust Xmin and Xmax to include the x-value. Then check that the y-value at that x also falls within Ymin and Ymax. The TI-84 does not scale automatically; you have to set the window so the whole tangent line is visible.

Radian vs. Degree Mode for Trig

Trigonometric functions are sensitive to mode. If you graph y = sin(x) with X from 0 to 6.28 in Radian mode, you see one full period. In Degree mode, the same window shows a nearly flat line. The slope of the tangent line will be wrong because the calculator interprets the x-value as degrees, not radians. Always match the mode to the problem: use Radian for calculus unless the problem explicitly uses degrees. Press [MODE] to toggle before you graph.

Non-Differentiable Points

The TI-84's nDeriv( feature does not check whether the derivative exists. At a corner like f(x) = |x| at x = 0, the calculator returns a slope of 0, which is wrong, the function is not differentiable there. If the problem involves a piecewise function, a cusp, or a vertical tangent, do not trust the calculator's numeric derivative. Compute the left-hand and right-hand limits of the difference quotient by hand. If they differ, there is no tangent line, and the calculator's answer is meaningless.

Vertical Tangents

A vertical tangent line has an undefined slope. The TI-84 cannot draw a vertical line through DRAW > Tangent( because it treats the equation as y = mx + b. If you encounter a vertical tangent, the calculator may draw a nearly vertical line with an extremely large slope, or it may fail entirely. The correct equation for a vertical tangent is x = a, which you must write yourself.

The single most practical step: after you get a slope from the TI-84, always verify the function is differentiable at that point by checking for corners, cusps, or vertical tangents. If the function is differentiable, use the point-slope form to write the equation by hand. If the problem asks for the normal line, take the negative reciprocal of the slope and use the same point. For linearization, the tangent line L(x) = f(a) + f'(a)(x-a) is the same line, use it to approximate values near a, but remember the error grows with the square of the distance from a. The most common mistake on the AP exam is writing the calculator's decimal slope without showing the exact derivative work. Always show the derivative rule, then use the calculator only to check your arithmetic.