Linear functions

Write, evaluate and interpret linear functions from words, tables and graphs, including slope and intercept in context. It is part of Algebra in the Math section. The book teaches it in Chapter 17, Linear Equations in Two Variables and Linear Functions, with 9 practice questions, and tests it 8 more times across the two full-length practice tests.

About Chapter 17

A linear function changes by the same amount every time its input increases by 1. Its graph is a straight line, and that line can be described by an equation in two variables, such as y = 2x + 5 or 3x + 4y = 20. The test splits this material into two skills, linear functions and linear equations in two variables, but they rest on the same few ideas: slope, intercepts and the forms of a line. This chapter teaches those ideas, then shows how the test asks about them.

What the question looks like

  • Find the equation. From two points, a table, a graph or a description, choose or build the equation of a line.
  • Interpret a number. "Which statement is the best interpretation of 1.5 in this context?" The number is usually a slope or an intercept.
  • Use function notation. Find f(4), or find the input that makes f(x) equal a given value.
  • Parallel and perpendicular lines. Find a line, a slope or a constant that makes two lines parallel or perpendicular.
  • Intercepts. Find where a line crosses an axis, or say what that crossing means in context.
  • Two-item combinations. An equation such as 12a + 8b = 240 describes all the ways to combine two items that cost $12 and $8 for a total of exactly $240.

How to recognize it

Two variables appear, x and y or two context variables, and neither is squared, under a radical or in an exponent. A function such as f(x) = 4 − 2x is linear because its input appears only to the first power. A table is linear when equal steps in x always produce equal steps in y. A graph is linear when it is a straight line.

A question that asks for a point satisfying two equations is a system (Chapter 18). A question about one line belongs here.

Common traps

  • Rise and run upside down: You divided the change in x by the change in y. Slope is rise over run.
  • Using a table's first row as the y-intercept: The first row's input is not 0. Work back to x = 0.
  • Sign errors in point-slope form: The point has a negative coordinate. Write x − (−3) as x + 3.
  • Misreading slope from standard form: The slope of Ax + By = C is −A/B, not A or A/B.
  • Calling the constant "the first unit's cost": The constant is the value when the input is 0.
  • Confusing f(a) with a: The question gives an output; set f(x) equal to it and solve.
  • Opposite or reciprocal, but not both: A perpendicular slope changes sign and flips. Check that the product is −1.
  • Treating a trade-off as a price difference: In ax + by = c, the slope is −a/b, a ratio.

Video lessons

Two short lessons from independent YouTube creators, chosen for this skill. The notes are ours; the videos belong to their creators, who are not affiliated with this site.

  • Slope and slope-intercept form (Gauth, 6:21). Watch for: Reading slope and intercept as a rate and a starting value.
  • An introduction to y = mx + b (Math with Mr. J, 14:16). Watch for: A refresher on slope-intercept form if linear functions feel shaky.

Practice this skill

The full method, worked examples and 9 practice questions with every answer choice explained are in Chapter 17 of the book and in the study workspace, where an AI coach can talk you through each one.

Independence

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