Precalculus Calculator

Four precalculus tools in one: a quadratic solver that shows the discriminant, roots (real or complex), vertex and axis of symmetry; a function grapher that plots and tabulates any expression in x; a unit-circle trig calculator with exact values for special angles; and an exponential and logarithm solver. Everything runs instantly in your browser.

What precalculus covers

Precalculus brings together algebra and trigonometry to prepare for calculus: functions and their graphs, polynomial and rational functions, exponential and logarithmic functions, trigonometric functions and the unit circle, identities, sequences, vectors and conic sections. The four tools here cover the calculations students meet most often and show intermediate values so you can check your own working.

Quadratic equations

For 2x² − 4x − 6 = 0, D = 16 + 48 = 64, so the roots are (4 ± 8) / 4: x = 3 and x = −1. The vertex is at x = 1, y = −8 — the minimum, because a is positive.

Graphing functions

Type any expression in x using + − * / and ^, brackets, and functions sin, cos, tan, asin, acos, atan, ln, log (base 10), sqrt, abs and exp, with constants pi and e. Implicit multiplication such as 2x or 3sin(x) works. The grapher plots the function, lists a table of values and finds approximate zeros where the graph crosses the x-axis. Trig functions use radians. For f(x) = x³ − 3x + 1 on [−3, 3], it finds three zeros near −1.879, 0.347 and 1.532.

The unit circle and exact trig values

For other angles, find the reference angle and use the quadrant to set the sign: “All Students Take Calculus” — all positive in quadrant I, sine in II, tangent in III, cosine in IV. The calculator shows exact values for multiples of 30° and 45° and decimals for every angle; in radians you can type expressions like 5pi/6.

Exponentials and logarithms

A logarithm answers “what power?”: log_b(x) = y means b^y = x. To solve an exponential equation like 2^x = 40, take logarithms: x = log₂(40) = ln 40 ÷ ln 2 ≈ 5.3219. The change-of-base formula lets you compute any logarithm with natural logs or base-10 logs. Key rules: log(xy) = log x + log y, log(x/y) = log x − log y, log(x^n) = n·log x.

Worked example: combining tools

A ball’s height is h(t) = −4.9t² + 14.7t + 2. In the quadratic tool with a = −4.9, b = 14.7, c = 2, the discriminant is 255.29, the positive root is about 3.13 seconds (when it hits the ground) and the vertex is at t = 1.5 s with a maximum height of about 13.03 m. Graphing −4.9x^2 + 14.7x + 2 from 0 to 3.5 shows the parabola and confirms the zero near 3.13.

Functions and transformations

Try it in the grapher: graph x^2, then (x-2)^2 + 1, and see the parabola shift right 2 and up 1. The same rules apply to every function family — trigonometric, exponential, logarithmic and polynomial.

Domain and range

The domain is the set of x-values a function accepts; the range is the set of outputs. Square roots need non-negative inputs, logarithms need positive inputs, and rational functions exclude values that make the denominator zero. When the grapher shows gaps, the function is undefined there — for example tan(x) at odd multiples of π/2 or ln(x) for x ≤ 0.

Using the calculator to study

Work each problem by hand first, then use the calculator to check. If your answer differs, compare the intermediate values — discriminant, vertex, reference angle or change-of-base step — to find where the difference starts. That habit catches sign errors and calculator-entry mistakes before a test.

Checking answers

• Substitute roots back into the original equation.
• Compare the vertex with the graph’s turning point.
• Use the unit circle to sanity-check signs of trig values in each quadrant.
• Remember that exponentials grow faster than any polynomial — check with the grapher.

Common mistakes

• Using degrees in the grapher when trig functions expect radians.
• Forgetting that ln and log are different bases.
• Dropping the ± in the quadratic formula.
• Writing −x^2 and expecting (−x)²: in the grapher, −x^2 means −(x²).
• Taking the log of zero or a negative number.

Privacy

All calculations run in your browser; nothing is stored.