A linear function graphs as a straight line. Plot the y-intercept, use slope to get a second point, and draw the line through both points. You can also create a table of x values and compute matching y values.
- A positive slope rises left to right, a negative slope falls.
- The y-intercept is where x = 0.
- Two correct points determine the line, but a third point helps catch mistakes.
Start from function form
In slope-intercept form y = mx + b, b is the y-intercept and m is the rate of change. Plot b first, then move by rise over run to locate more points.
For example, if m = -2/3, move down 2 and right 3 from the intercept. Equivalent moves such as up 2 and left 3 also work.
Read meaning from the graph
In applications, slope represents how much y changes for each 1-unit change in x. The intercept is the starting value when x is zero. Label axes with units so the graph communicates a real statement.
- Check scales on both axes before plotting.
- Use consistent spacing for each grid step.
- Interpret slope and intercept in words, not symbols only.
Graph a taxi fare model
Fare y in dollars follows y = 2.5x + 4, where x is miles traveled.
- Plot the y-intercept at (0, 4).
- Use slope 2.5 = 5/2: from (0, 4), move right 2 and up 5 to (2, 9).
- Plot a third point by substitution, such as x = 4 gives y = 14.
- Draw the line through the points and label axes with units.
Common mistakes
- Swapping x and y coordinates when plotting points.
- Reading slope as run over rise instead of rise over run.
- Using uneven axis intervals that distort the line.
- Forgetting units, so the graph loses context.
Try one
For y = -3x + 2, what two points can you plot quickly?
Start with intercept (0, 2). Using slope -3, move right 1 and down 3 to get (1, -1).
Sources
- OpenStax: College AlgebraOpen textbook with full chapter coverage, worked examples, and practice problems.
- Khan Academy: Algebra and Algebra 2Official lessons and practice sets for core algebra skills.