What is a table of values (function table)?

A table of values — also called a function table, input-output table or x-y table — lists input values (x) next to the output values (y or f(x)) a rule produces. Each column or row is one ordered pair you can plot. For y = 2x + 1 and x = −2, −1, 0, 1, 2, the outputs are −3, −1, 1, 3, 5, giving the points (−2, −3), (−1, −1), (0, 1), (1, 3), (2, 5).

How do you complete a function table?

Substitute each input into the rule and simplify using the order of operations. For y = −x² + 1 with x = −2, −1, 0, 1: y = −(−2)² + 1 = −3; y = −(−1)² + 1 = 0; y = −(0)² + 1 = 1; y = −(1)² + 1 = 0. Note that −x² means −(x²), so x = −2 gives −4 + 1 = −3, not 5.

How do I fill in a table using a function rule such as y = 3 + 2x?

Replace x with each input. For y = 3 + 2x and x = −2, −1, 0, 1, 2: y = −1, 1, 3, 5, 7. Every time x goes up by 1, y goes up by 2 (the coefficient of x), which is a quick way to check your table.

How do you find the function from a table of values?

Check the differences. If the outputs change by the same amount for equal input steps, the rule is linear: y = mx + b, where m = change in y ÷ change in x and b is the output at x = 0. For x = 0, 1, 2, 3 and y = 9, 14, 19, 24, each step adds 5 and y = 9 at x = 0, so y = 5x + 9 (answer choice a). Constant second differences mean quadratic; a constant ratio means exponential. Use the "Table → rule" tab to do this automatically.

Inputs 4, 6, 7 give outputs 16, 24, 28. What is the output when the input is n?

4n. Each output is 4 times its input (16 ÷ 4 = 24 ÷ 6 = 28 ÷ 7 = 4), and the line passes through the origin, so the rule is output = 4 × input — this is direct variation. With input n, the output is 4n.

What are the table values for y = x² − 3?

For x = −3, −2, −1, 0, 1, 2, 3: y = 6, 1, −2, −3, −2, 1, 6. The lowest output is −3 at x = 0 (the vertex), and the values are symmetric about x = 0. The zeros are x = ±√3 ≈ ±1.732, between the table rows where y changes sign.

What is the table of values for y = x² + x?

For x = −3, −2, −1, 0, 1, 2: y = 6, 2, 0, 0, 2, 6. The outputs are 0 at x = −1 and x = 0 (the zeros), and the vertex is halfway between them at x = −1/2, where y = −1/4. Second differences are all 2, confirming a quadratic.

Complete the function table for y = −5x − 1 with inputs −2, −1, 0, 1.

y = −5(−2) − 1 = 9; y = −5(−1) − 1 = 4; y = −5(0) − 1 = −1; y = −5(1) − 1 = −6. The outputs 9, 4, −1, −6 drop by 5 each step because the slope is −5.

How do I make a table of x and y values for the line y = 8x + 4?

Choose a few x-values on both sides of 0. For x = −2, −1, 0, 1, 2: y = −12, −4, 4, 12, 20. The y-intercept is (0, 4), and each step of 1 in x adds 8 to y. Two points are enough to draw the line; a third checks your arithmetic.

How do you make a table of values for x + y = 5?

Solve for y first: y = 5 − x. Then substitute: x = −2, −1, 0, 1, 2, 3 gives y = 7, 6, 5, 4, 3, 2. Every pair adds to 5. The same method gives x + y = 6 → y = −x + 6 (8, 7, 6, 5, 4, 3) and x + y = 0 → y = −x (2, 1, 0, −1, −2, −3). The calculator solves for y automatically when you type the equation.

Copy and complete the table for output = (10 × input) − 3 and output = (3 × input) + 1.

For inputs 1, 2, 3, 4, 5: output = (10 × input) − 3 gives 7, 17, 27, 37, 47, and output = (3 × input) + 1 gives 4, 7, 10, 13, 16. Multiply first, then add or subtract (order of operations). Type the rule with the words "input" and "output" and the calculator keeps them as column headings.

How do I create a table for h(t) = −2t² + 8t?

Substitute t = 0, 1, 2, 3, 4: h = 0, 6, 8, 6, 0. The maximum is 8 at t = 2, and h = 0 at t = 0 and t = 4. If the rule really is h(t) = −2t + 8t (no square), it simplifies to 6t and the outputs are 0, 6, 12, 18, 24.

A table has outputs 1, 16, 64, 256, 1,024 for x = 1 to 5. What does its graph look like?

After the first value, each output is 4 times the previous one (16 → 64 → 256 → 1,024), so the data grow exponentially: the graph starts flat and curves steeply upward. It is not a straight line (the differences are not constant) and not a parabola (the second differences are not constant). The pure pattern y = 4ˣ would give 4, 16, 64, 256, 1,024.

How do I tell if a table represents a linear function?

Check that x goes up in equal steps and y changes by the same amount every step. y = 2x + 1 at x = 0, 1, 2, 3, 4 gives 1, 3, 5, 7, 9 — always +2, so it is linear. y = x² gives 0, 1, 4, 9, 16 — differences 1, 3, 5, 7 grow, so it is not linear. If the x-steps are uneven, compare (change in y) ÷ (change in x) instead.

How do I make a table of values on a graphing calculator (TI-84 or Desmos)?

TI-84 Plus: press Y=, type the rule into Y1, press 2nd WINDOW (TBLSET) to set TblStart (first x) and ΔTbl (step), then press 2nd GRAPH (TABLE). Set Indpnt to Ask to type your own x-values. Desmos: type the rule in function notation, such as f(x) = 2x + 1, click Edit List, then Create Table; it starts with x from −2 to 2 and you can add rows.

Which x-values should I choose for a table of values?

Use about 5 to 7 inputs that include 0 and values on both sides (−3 to 3 is typical). For a parabola, center the table on the vertex x = −b/(2a). Avoid or flag inputs that make the rule undefined: y = 1/x at x = −2, −1, 0, 1, 2 gives −1/2, −1, undefined, 1, 1/2. For √x start at 0, and for trigonometric functions use steps of 30° or π/6.

How do I make a table of values from 1 to 10?

Set start = 1, end = 10 and step = 1 in the calculator. For y = x², the outputs are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Use a step of 0.5 to get 19 rows, or switch to "Specific values" to enter any list.

Is every table of values a function?

Only if each input has exactly one output. A table with x = 2 paired with both 4 and 5 is a relation but not a function. Different inputs may share an output (y = x² gives 4 for both x = −2 and x = 2), which is allowed. The rule finder flags tables where an input repeats with different outputs.

What is the difference between an input-output table, an x-y table and a function table?

Nothing important — they are three names for the same idea. Elementary textbooks say input and output (or "In" and "Out"), algebra courses say x and y, and function notation writes x and f(x). A "T-chart" is the same table drawn as two columns.

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Function table · input-output table · x-y table

Table of Values Calculator (Function Table)

Quick answer

To make a table of values, pick input values for x, substitute each one into the rule, and simplify to get y. Each row is an ordered pair (x, y). For y = 2x + 1 with x = −2, −1, 0, 1, 2, the outputs are −3, −1, 1, 3, 5 — they rise by 2 each step, so the function is linear with slope 2. To go the other way, look at the differences: constant first differences mean linear, constant second differences mean quadratic, and a constant ratio means exponential.

Type any rule — 2x + 1, y = x^2 − 3, x + y = 5, h(t) = −2t² + 8t or output = 10 × input − 3 — and choose a range or the exact inputs from your worksheet. You get the completed table, the substitution for every row, differences, zeros and a graph. Paste a table into the second tab to find its rule.

Real expression engine: powers, roots, |x|, fractions, trig, logs, eˣ
Solves equations like x + y = 5 or 2x + 3y = 6 for y first
Show-work lines such as y = 2(−2) + 1 = −3
Finds the rule from a table and fills in blanks (including input n)

Last updated October 1, 2026. Methods follow OpenStax College Algebra 2e and Elementary Algebra 2e. · complete a function table · find the rule from a table

Table of values calculator

Type a function or equation to fill in the table — or paste a table to find its rule.

Function, rule or equation

Examples: 2x+1, y = x^2 - 3, x + y = 5, h(t) = -2t^2 + 8t, output = 10 × input − 3, sqrt(x), |x|, 2^x

Input values

Start

End

Step

Show outputs as

Show differences

Rule

y = 2x + 1

Linear7 rowsincreasing

x	y	1st diff	2nd diff
−3	−5
−2	−3	2
−1	−1	2	0
0	1	2	0
1	3	2	0
2	5	2	0
3	7	2	0

y-intercept (x = 0)

Zeros in this range

−0.5

Lowest output in table

−5 at x = −3

Highest output in table

7 at x = 3

First differences are all 2 → linear. Rate of change (slope) = 2 ÷ 1 = 2.

Show work (substitution)

y = 2(−3) + 1 = −5
y = 2(−2) + 1 = −3
y = 2(−1) + 1 = −1
y = 2(0) + 1 = 1
y = 2(1) + 1 = 3
y = 2(2) + 1 = 5
y = 2(3) + 1 = 7

Table of values for common functions (x = −3 to 3)

Every value below is computed by the same engine as the calculator. “undefined” marks inputs outside the domain.

Function	x = −3	x = −2	x = −1	x = 0	x = 1	x = 2	x = 3	Pattern
y = x	−3	−2	−1	0	1	2	3	Identity
y = 2x + 1	−5	−3	−1	1	3	5	7	Linear, slope 2
y = x²	9	4	1	0	1	4	9	Quadratic (parabola)
y = x² − 3	6	1	−2	−3	−2	1	6	Parabola shifted down 3
y = −x² + 1	−8	−3	0	1	0	−3	−8	Opens downward
y = x³	−27	−8	−1	0	1	8	27	Cubic
y = \|x\|	3	2	1	0	1	2	3	Absolute value (V shape)
y = 2^x	1/8	1/4	1/2	1	2	4	8	Exponential growth
y = 1/x	−1/3	−1/2	−1	undef.	1	1/2	1/3	Reciprocal — undefined at 0
y = √x	undef.	undef.	undef.	0	1	1.414	1.732	Square root — undefined below 0

How to complete a function table

A function table is finished one input at a time. Write the rule, put each input in brackets where x appears, and simplify. Brackets matter most with negative inputs and squares.

1. Write the rule as y = …
If the equation is in another form, such as x + y = 5, solve for y first: y = 5 − x.
2. Choose the inputs
Use the x-values given, or pick 5 to 7 values including 0 and numbers on both sides, such as −3 to 3.
3. Substitute each input
Replace x with the value in brackets, e.g. y = 2(−2) + 1, so negative signs are handled correctly.
4. Simplify with the order of operations
Powers first, then multiplication and division, then addition and subtraction. −x² at x = −2 is −4.
5. Record and check the pairs
Write each (x, y) pair. Equal differences mean linear; equal second differences mean quadratic; a constant ratio means exponential.

Worked example: complete the table for y = −x² + 1

Inputs −2, −1, 0, 1

y = −(−2)² + 1 = −3
y = −(−1)² + 1 = 0
y = −(0)² + 1 = 1
y = −(1)² + 1 = 0

x	−2	−1	0	1
y	−3	0	1	0

Common mistake: −x² is −(x²). Squaring first gives (−2)² = 4, then the minus sign makes it −4. Typing (-x)^2 + 1 would give 5 instead.

Worked example: equation in standard form

2x + 3y = 6 — solve for y, then substitute

Subtract 2x and divide by 3: y = −(2/3)x + 2. Choosing multiples of 3 for x keeps the outputs whole numbers:

y = −(2/3)(−3) + 2 = 4
y = −(2/3)(0) + 2 = 2
y = −(2/3)(3) + 2 = 0

x	−3	0	3
y	4	2	0

The same steps turn x + y = 5 into y = −x + 5, so x = −2, −1, 0, 1, 2, 3 gives y = 7, 6, 5, 4, 3, 2.

Completed function tables for common homework rules

Check your answers against these tables, or load any rule into the calculator to see the steps.

y = x² − 3

x	−3	−2	−1	0	1	2	3
y	6	1	−2	−3	−2	1	6

Vertex (0, −3); symmetric outputs.

y = x² + x

x	−3	−2	−1	0	1	2
y	6	2	0	0	2	6

Zeros at x = −1 and x = 0.

y = −x² + 1

x	−2	−1	0	1
y	−3	0	1	0

−(−2)² + 1 = −4 + 1 = −3.

y = −5x − 1

x	−2	−1	0	1
y	9	4	−1	−6

Outputs fall by 5 each step.

y = 8x + 4

x	−2	−1	0	1	2
y	−12	−4	4	12	20

y-intercept 4, slope 8.

y = 3 + 2x

x	−2	−1	0	1	2
y	−1	1	3	5	7

Same as y = 2x + 3.

output = (10 × input) − 3

input	1	2	3	4	5
output	7	17	27	37	47

Multiply first, then subtract.

output = (3 × input) + 1

input	1	2	3	4	5
output	4	7	10	13	16

Outputs rise by 3.

y = x (y = 1x)

x	−2	−1	0	1	2
y	−2	−1	0	1	2

Every output equals its input.

h(t) = −2t² + 8t

t	0	1	2	3	4
h(t)	0	6	8	6	0

Peak height 8 at t = 2.

y = x + 2 at x = −9, −6, 8

x	−9	−6	8
y	−7	−4	10

Any inputs work — not just −3 to 3.

y = sin x (degrees)

x	0	30	60	90	120	150	180
y	0	1/2	0.866025	1	0.866025	1/2	0

Exact values shown as decimals.

How to find the function rule from a table of values

Working backwards from a table is the same skill as “which equation represents the function shown in the table?” Look at how the outputs change when the inputs change by equal steps:

Linear: y = mx + b

Same first difference every step. Slope m = Δy ÷ Δx; b is the output at x = 0 (or b = y − mx from any point).

Quadratic: y = ax² + bx + c

First differences change but second differences are constant. With x-steps of 1, a = (second difference) ÷ 2.

Exponential: y = a · bˣ

Each output is the previous one times the same number b. a is the output at x = 0.

Inputs (x)	Outputs (y)	Rule found	Missing output	How
0, 1, 2, 3	9, 14, 19, 24	y = 5x + 9	—	Rate of change: (14 − 9) ÷ (1 − 0) = 5 — the same between every pair of points, so the table is linear.
4, 6, 7, n	16, 24, 28, ?	y = 4x	n → 4n	Rate of change: (24 − 16) ÷ (6 − 4) = 4 — the same between every pair of points, so the table is linear.
2, 5, 9	7, 16, 28	y = 3x + 1	—	Rate of change: (16 − 7) ÷ (5 − 2) = 3 — the same between every pair of points, so the table is linear.
1, 2, 3, 4, 5	1, 4, 9, 16, ?	y = x²	5 → 25	First differences change, so the rule is not linear.
0, 1, 2, 3, 4	3, 6, 12, 24, ?	y = 3 · 2^x	4 → 48	Differences are not constant, but the ratio 6 ÷ 3 = 2 repeats → exponential.

The third row has uneven inputs (2, 5, 9), so compare Δy ÷ Δx rather than raw differences: 9 ÷ 3 = 12 ÷ 4 = 3, giving y = 3x + 1. Examples in the calculator's rule tab: Linear: 9, 14, 19, 24; Input n → output?; Quadratic: 1, 4, 9, 16; Exponential: 3, 6, 12, 24.

Linear vs quadratic vs exponential tables

The same inputs give very different patterns. The difference and ratio columns are what identify the function type.

x	y = 2x + 1	1st diff	y = x²	1st diff	2nd diff	y = 2ˣ	ratio
0	1		0			1
1	3	+2	1	+1		2	×2
2	5	+2	4	+3	+2	4	×2
3	7	+2	9	+5	+2	8	×2
4	9	+2	16	+7	+2	16	×2

Linear: First differences are all 2 → linear. Rate of change (slope) = 2 ÷ 1 = 2.
Quadratic: Second differences are all 2 → quadratic (parabola), leading coefficient a = 1, opening up.
Exponential: Each output is 2 × the one before (constant ratio) → exponential, base b = 2 per 1 unit of x.

Choosing good x-values

So the table shows the important features of the graph

Include x = 0 — its output is the y-intercept.
Use negatives and positives (−3 to 3 or −2 to 2) so you see both sides of the graph.
Center a parabola on its vertex x = −b/(2a). For y = x² + x the vertex is at x = −1/2, so −3 to 2 is balanced: 6, 2, 0, 0, 2, 6.
Watch for undefined points: y = 1/x at x = −2, −1, 0, 1, 2 → −1/2, −1, undefined, 1, 1/2. Start √x at 0.
For fractions in the rule (y = (2/3)x), pick multiples of the denominator so outputs are whole numbers.
For sine and cosine, use 30° or π/6 steps: sin x at 0, 30, 60, 90, 120, 150, 180° → 0, 0.5, 0.866, 1, 0.866, 0.5, 0.

Making a table on a graphing calculator

TI-84 Plus and Desmos

TI-83 / TI-84 Plus

Press Y= and type the rule into Y1 (use X,T,θ,n for x).
Press 2nd WINDOW (TBLSET). Set TblStart (first x) and ΔTbl (step).
Leave Indpnt: Auto for a range, or choose Ask to type specific x-values.
Press 2nd GRAPH (TABLE) and scroll with the arrow keys.

Desmos

Type the rule in function notation, e.g. f(x) = 2x + 1.
Click Edit List, then Create Table next to the expression.
Desmos fills x = −2 to 2; edit or add x-values to evaluate more inputs.

This calculator does the same without setup, accepts equations not solved for y, and shows the substitution for each row.

How to type a rule

Powers: x^2, x² or x2; multiplication can be implied: 3(x−1)^2, (x+1)(x−1)
Any prefix works: y =, f(x) =, h(t) =, output = (with input as the variable)
Equations with x and y: x + y = 5, 2x + 3y = 6, y − x² = 3, xy = 12
Functions: sqrt(x) or √x, |x|, 2^x, e^x, ln(x), log(x), sin(x)
Fractions: 1/2x + 3 means (1/2)x + 3; use 1/(2x) for one over 2x
Inputs list: -2, -1, 0, 1/2, pi — switch “Input values” to Specific values