Function Table & Graphing Table Maker

A function table lists inputs in one column and the outputs a rule gives them in the other. You will also see it called an input-output table or an x and f(x) table. Each row is one input and its one output.

Think of the rule as a machine. For f(x) = 2x + 1, the machine doubles whatever goes in and adds 1. Put in 2, double it to 4, add 1, and 5 comes out. Put in −2 and −3 comes out.

Open this table in the Function Lab

Filling in a function table is substitution, done once per row. Here is the method with f(x) = 3x − 4.

1. Find x in the rule. In f(x) = 3x − 4, x is multiplied by 3, then 4 is subtracted.
2. Replace x with the first input, in parentheses: f(2) = 3(2) − 4.
3. Follow the order of operations, multiplying before you subtract: 3(2) − 4 = 6 − 4 = 2.
4. Write the output in the same row, then repeat for every input.
5. Check the pattern down the column. In a linear table the outputs change by the same amount each time x goes up by 1, so an output that breaks the pattern is usually an arithmetic slip.

Check your table in the Function Lab

Sometimes the table comes first and you have to work out the rule. For a linear table, you only need two numbers: how fast the outputs change and where they start.

1. Check that x goes up by the same amount each row. Here it goes up by 1.
2. Subtract each output from the next: 8 − 5 = 3, 11 − 8 = 3, 14 − 11 = 3. The output rises by 3 every time, so 3 is the rate of change, the number that multiplies x.
3. Find the output when x = 0. It is 5, the starting value, the number you add.
4. Put them together: f(x) = 3x + 5.
5. Test the rule on a row you did not use: f(4) = 3(4) + 5 = 17. It matches.

Test f(x) = 3x + 5 in the Function Lab

A linear table changes by the same amount at every step, and its graph is a straight line. Every table above is linear.

The table for f(x) = x² is different. Its outputs change by −3, then −1, then +1, then +3, so it is nonlinear and its graph bends into a U-shaped parabola. In Algebra 1 you will notice that those changes themselves go up by 2 each time; a constant second difference is the sign of a quadratic.

Graph f(x) = x² in the Function Lab

A table is a function when every input has exactly one output. Scan the x column: if the same x appears twice with two different outputs, the table is not a function. In the table below, x = 2 gives both 5 and 7.

Repeated outputs are fine. In the x² table, both −2 and 2 give 4, and it is still a function, because each input has only one output.

On a graph, two outputs for one input are two points stacked on the same vertical line, which is why the vertical line test works. The Function Lab checks this for you: enter the same x twice with different outputs and it warns that the table is not a function.

A graphing table is just a function table read as points. Each row becomes an ordered pair (x, f(x)): the f(x) = 2x + 1 table gives (−2, −3), (−1, −1), (0, 1), (1, 3) and (2, 5).

1. Plot each point: start at the origin, move across to x, then up or down to f(x).
2. If the inputs can be any number, such as time or distance, connect the points with a line or smooth curve.
3. If the inputs can only be whole things, such as tickets or people, leave the points unconnected.
4. Look for a point that does not fit. All five points above sit on one straight line; a point off the line means that row has an error.

Plot the 2x + 1 table

A taxi charges $1.25 to start the ride plus $2.50 per mile. The cost for x miles is f(x) = 2.5x + 1.25.

The cost goes up by $2.50 for every mile; that is the rate of change. The x = 0 row, $1.25, is the starting fee; on the graph it is the y-intercept. Every linear table hides those same two numbers, which is why finding them is the key to writing the rule.

Graph the taxi fares