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Algebra I to AP Precalculus · Module 4 of 14

Linear Functions & Graphing

Function notation, slope, the three forms of a line, and writing an equation from given information.

In this module
  1. Relations, Functions & Function Notation
  2. Slope & Rate of Change
  3. Graphing Lines (Three Forms)
  4. Writing Equations of Lines
  5. Check yourself

Relations, Functions & Function Notation

  • A function assigns exactly one output to each input; the vertical line test checks this on a graph.
  • f(x) means "the output of function f for input x" — f(3) means substitute 3 for x.
  • Domain is the set of valid inputs (x-values); range is the set of resulting outputs (y-values).
Worked example

If f(x) = 2x − 5, find f(4).

Solution. Substitute x = 4: f(4) = 2(4) − 5 = 8 − 5 = 3.

Slope & Rate of Change

  • Slope m = (y2 − y1) / (x2 − x1) = rise/run between any two points on a line.
  • Positive slope rises left to right; negative falls; zero is horizontal; undefined is vertical.
  • Slope represents a constant rate of change — e.g. dollars per hour, distance per second.
Slope as rise over run
xyy = 2x + 1run 1rise 2y-intercept (0, 1)
y = 2x + 1: from any point, moving right 1 means moving up 2, so the slope is 2. The line crosses the y-axis at 1.
Worked example

Find the slope through (2, 3) and (6, 11).

Solution. m = (11 − 3) / (6 − 2) = 8/4 = 2.

Graphing Lines (Three Forms)

  • Slope-intercept y = mx + b: plot the y-intercept b, then use slope m as rise/run for a second point.
  • Standard form Ax + By = C: find the x-intercept (set y = 0) and y-intercept (set x = 0), then connect them.
  • Point-slope y − y1 = m(x − x1): built directly from one point and the slope.
Worked example

Graph y = −2x + 3 using slope and intercept.

Solution. Plot the y-intercept (0, 3). Slope −2 = −2/1, so from (0, 3) move right 1, down 2 to (1, 1); draw the line through both points.

Writing Equations of Lines

  • Given a slope and a point, plug into point-slope form y − y1 = m(x − x1), then simplify if needed.
  • Given two points, find the slope first, then use point-slope form with either point.
  • Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals (they multiply to −1).
Worked example

Write the equation of the line through (1, 2) parallel to y = 3x − 4.

Solution. Parallel lines share slope, so m = 3. Point-slope: y − 2 = 3(x − 1). Simplify: y = 3x − 1.

Check yourself

2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.

1. If f(x) = 3x + 1, find f(−2).

2. Find the slope through (0, −3) and (4, 5).

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