Why this chapter matters
Linear functions are the first major function family and a foundation for comparing rates, reading trends, and choosing useful models.
What you will learn
- Calculate and interpret slope as a rate of change.
- Graph a line from an equation, table, or two points.
- Write equations in slope-intercept and point-slope forms.
Understand the core ideas
A linear function has a constant rate of change, so equal changes in x create equal changes in y. That constant rate is slope, computed as change in output divided by change in input. If slope is positive, the graph rises as you move right; if negative, it falls. The y-intercept gives the output when input is zero, so it is often the starting amount in a context like cost, distance, or temperature. Units matter: a slope of 3 might mean 3 dollars per hour or 3 miles per minute depending on the model.
You can represent the same line as an equation, a table, or a graph. Slope-intercept form y = mx + b is efficient for reading rate and start quickly, while point-slope form y - y1 = m(x - x1) is efficient when one point and slope are known. To compare relationships, place both lines on one graph and inspect slope, intercept, and intersection. The intersection solves the question where the two models produce the same output.
Key terms
- slope
- The constant rate of change of a line, often written as rise over run.
- y-intercept
- The point where a graph crosses the y-axis, with x = 0.
- slope-intercept form
- Equation form y = mx + b, where m is slope and b is the y-intercept.
- point-slope form
- Equation form y - y1 = m(x - x1) using slope and one known point.
Write a line from two points
Find the equation of the line through (2, 7) and (6, -1).
- Compute slope: m = (-1 - 7) / (6 - 2) = -8/4 = -2.
- Use point-slope form with point (2, 7): y - 7 = -2(x - 2).
- Distribute on the right side: y - 7 = -2x + 4.
- Add 7 to both sides to write slope-intercept form: y = -2x + 11.
A common misconception
Claim: A steeper line always means a larger y-intercept.
Correction: Steepness is controlled by slope, while intercept is a separate value. A line can be steep and still cross the y-axis low or even below zero.
Lessons in this chapter
- Graphing linear functionsPlot lines from equations and tables. Read the full guide →
- Slope and rate of changeCalculate rise over run and interpret units.
- Slope-intercept formUse y = mx + b to identify rate and starting value. Read the full guide →
- Writing a line from dataBuild a linear equation from points or a real situation.
Study task
Chapter checkpoint
What are the slope and y-intercept of y = -3x + 5?
The slope is -3 and the y-intercept is 5, so the line passes through (0, 5).