StudyPebble

Finding a Linear Function From an SAT Table

Instead of an equation, the question gives you a small table of inputs and outputs and asks for the function, for a missing value, or for which equation the table represents. The path is always the same two steps: find the rate of change between rows, then back out the value at an input of zero.

The rule

Divide the change in output by the change in input to get the rate, then substitute one row to find the value at an input of zero.

Say the table shows inputs 1, 2, 3 and 4 with outputs 9, 14, 19 and 24. The inputs go up by 1 each row and the outputs go up by 5 each row, so the rate of change is 5 per unit. Because the change is constant, the relationship is linear and the function has the form f of x equals 5x plus b.

To find b, take any single row and substitute. Using x equal to 1 and output 9 gives 9 equals 5 plus b, so b equals 4 and the function is f of x equals 5x plus 4. Check with a second row to be safe: at x equal to 3 the rule gives 15 plus 4, which is 19, matching the table. That check costs a few seconds and catches arithmetic slips.

Uneven input steps are the version that trips people. If the inputs are 2, 5 and 9 with outputs 13, 22 and 34, you cannot read the rate off the output column directly. Divide the change in output by the change in input: from x equal to 2 to x equal to 5, the output rises 9 while the input rises 3, so the rate is 3 per unit. Confirm with the next pair, where the output rises 12 as the input rises 4, which is also 3.

That division is also how the SAT tests whether a table is linear at all. Compute the ratio between at least two different pairs of rows. If all the ratios agree, the relationship is linear. If the outputs instead multiply by a constant factor each step, for example 3, 6, 12, 24, the relationship is exponential and no linear function fits it.

Some tables never ask for the full equation. They ask for the output at an input the table does not list, or for the input that gives a certain output. Once you have the rate and the value at zero you can answer either, but you can also often extend the pattern directly. If the rate is 5 per unit and you know the output at x equal to 4 is 24, then the output at x equal to 7 is 24 plus 3 times 5, which is 39.

The trap

Reading the rate straight off the output column when the inputs do not increase by exactly 1 each row.

Try it

Try one
Real question

The cost to rent a bike is $5 plus $3 per hour. After how many hours does the cost equal $26?

A city's recycling center processed 8,400 tons in its 3rd year of operation and 14,700 tons in its 10th year. Tonnage has grown at a constant rate. How many tons does the model predict for the 16th year?

  • A) 14,400
  • B) 19,200
  • C) 20,100 ✓
  • D) 21,000

Why: Tonnage rose 6,300 over 7 years, which is 900 tons per year. From year 10 to year 16 is 6 years, so add 5,400: 14,700 + 5,400 = 20,100. Choice D counts 7 years instead of 6. Answering 14,400 means you multiplied the annual growth by 16 and ignored the model's starting value. Answering 19,200 means you counted 12 years of growth between the 3rd and 16th years instead of 13. Answering 21,000 means you counted 7 years of growth after the 10th year instead of 6.

Practice more of these free

Questions students ask

How can I tell from a table whether the relationship is linear?

Check whether the change in output divided by the change in input is the same between every pair of consecutive rows. Constant differences per unit of input means linear. If the outputs are instead being multiplied by the same number each step, the relationship is exponential.

Do I have to use the first row to find the y-intercept?

No, any row works. Substitute the rate and one pair of values into the equation and solve for the constant. Using a row with small numbers or an input close to zero usually keeps the arithmetic simplest.

Guides that cover this

More on linear functions