A table of values is the most direct way to understand a function: pick inputs, compute outputs, and look for patterns. This calculator builds the table for any function of x, adds first and second differences, recognizes linear, quadratic and exponential patterns, flags sign changes and undefined points, and plots the curve with the tabulated points marked.
How to use the function table calculator
- Type the function, for example
x^2 - 2x - 3,2^x,sqrt(x)orsin(x). - Enter the start x, end x and step. Expressions such as
pi/4work for all three. - Keep Show first and second differences checked if you want the pattern columns.
- Read the table below the calculator. The tape summarizes the number of rows, the largest and smallest values and any zeros; the graph shows the curve.
How a function table is built
Starting from x0, each row adds the step h:
The difference columns compare neighboring rows:
For a polynomial of degree n, the n-th differences are constant and equal n! × (leading coefficient) × hn. For a quadratic ax2 + … with step h, the second difference is always 2ah2.
Worked example
Tabulate f(x) = x2 − 2x − 3 from −2 to 4 with step 1.
| x | f(x) | Δf | Δ²f |
|---|---|---|---|
| −2 | 5 | ||
| −1 | 0 | −5 | |
| 0 | −3 | −3 | 2 |
| 1 | −4 | −1 | 2 |
| 2 | −3 | 1 | 2 |
| 3 | 0 | 3 | 2 |
| 4 | 5 | 5 | 2 |
Zeros: f(−1) = 0 and f(3) = 0, so the graph crosses the x-axis at −1 and 3, matching the factored form (x + 1)(x − 3).
Minimum in the table: −4 at x = 1, the vertex of the parabola.
Pattern: the second differences are all 2 = 2 × 1 × 1², confirming a quadratic with leading coefficient 1.
Symmetry: the values mirror around x = 1 (−3 at 0 and 2, 0 at −1 and 3).
Reading patterns in a table
| What you see | What it suggests |
|---|---|
| Constant first differences | linear function, slope = difference ÷ step |
| Constant second differences | quadratic function |
| Constant ratio yk/yk−1 | exponential function |
| Values repeat after a fixed number of rows | periodic function, such as sine or cosine |
| Sign change between rows | a zero (or a break) lies between them |
Sign changes are a practical way to locate roots: if f(2) is negative and f(3) is positive for a continuous function, a root lies in between. Shrinking the step around that interval narrows it down, the same idea as the bisection method used by numerical solvers.
A sign change can also come from a vertical asymptote rather than a root. For 1/(x − 1), the values jump from negative to positive around x = 1 without ever passing through zero, which is why the calculator says “a zero or a break.”
Related tools
To find the exact roots of a quadratic, use the quadratic formula calculator. The polynomial derivative calculator gives the slope at any row, and trigonometric function graphs explore sine and cosine with amplitude and phase controls. To estimate values between rows, use linear interpolation.
Frequently asked questions
How many rows can the table have?
Up to 201 rows. If your range and step would produce more, increase the step or shorten the range.
What are first and second differences for?
They reveal the type of function from equally spaced values. Constant first differences mean the function is linear; constant second differences mean it is quadratic. A constant ratio between consecutive values points to an exponential function.
Why does a row say undefined?
The function has no real value at that x, such as sqrt(−1), ln(0) or 1/(x − 1) at x = 1. Infinite values from division by zero show as ∞ or −∞.
Can the step be a fraction of π?
Yes. Start, end and step accept expressions such as pi/6 or 0.25, which is handy for tabulating trigonometric functions at standard angles. Trig functions use radians.