Convert between representations (equation, table, graph, verbal description). | Linear functions | SAT Math

A linear function can be shown as an equation, table, graph, or word description with the same slope and starting value.

Core Idea

A linear function can be expressed as an equation, a table, a graph, or a verbal description — all carry the same two pieces of information: the slope and a point (often the y-intercept).

Understanding

Translate one feature at a time. A linear representation always carries the same two pieces of information: slope and starting value.

Equation → table: pick x-values and compute y.
Equation → graph: plot the y-intercept, then use the slope to get a second point.
Table → equation: find the slope from two rows, then solve for $𝑏$ .
Graph → equation: read the y-intercept, then count rise/run between two grid points.
Verbal → equation: identify the rate of change and the starting value from the description.

The SAT often mixes these forms in one question, so translate the known features into the target form and check one second point before you commit.

Step by Step

From the given representation, identify the slope (rate of change) and one known point or the y-intercept.
Translate into the target form: plug into $𝑦 = 𝑚 𝑥 + 𝑏$ for an equation, build rows for a table, or plot points for a graph.
Verify with a second data point — substitute back in to confirm the equation, table row, or graph point matches.

Misconceptions

Assuming every table starts at $𝑥 = 0$ — if it doesn't, you need to calculate $𝑏$ separately rather than just reading the first $𝑦$ -value.
Matching a graph to an equation by checking only the y-intercept and ignoring the slope (or vice versa). Always verify both.
Misreading graph scales — if gridlines represent 2 units instead of 1, the slope calculation is off by a factor of 2.

Question

Worked Example

A table shows the following values for a linear function: when $𝑥 = 1$ , $𝑦 = 7$ ; when $𝑥 = 3$ , $𝑦 = 13$ ; when $𝑥 = 5$ , $𝑦 = 19$ . Which equation represents this function?

Select an answer to see the explanation