Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 1 linear functions? Then this Algebra 1 lesson on Writing and Graphing Linear Functions in Three Forms explains each idea step by step, with worked examples.
Quick answer: A linear function has a constant rate of change called slope: the change in y divided by the change in x. It can be written as y = mx + b, y − y₁ = m(x − x₁) or Ax + By = C. Parallel lines share a slope, and perpendicular slopes are negative reciprocals.
Standards: F-IF.C.7a, F-LE.A.1a, F-LE.A.2, A-CED.A.2 · Level: Grade 9 Algebra 1
How do you find the slope of a line?
Slope measures steepness: the change in y divided by the change in x between two points.
m = y2 − y1x2 − x1
What are the three forms of a linear equation?
| Form | Equation | Best used when |
|---|---|---|
| Slope-intercept | y = mx + b | you know the slope and the y-intercept |
| Point-slope | y − y1 = m(x − x1) | you know the slope and any point |
| Standard | Ax + By = C | you want both intercepts quickly |
- Slope: (7 − 1) ÷ (4 − 1) = 2.
- Point-slope with (1, 1): y − 1 = 2(x − 1).
- Distribute and solve for y: y = 2x − 1 (slope-intercept form).
- Standard form: 2x − y = 1.
How do you graph from standard form?
Set x = 0 to find the y-intercept and y = 0 to find the x-intercept, then draw the line through both points. For 2x + 3y = 12 the intercepts are (6, 0) and (0, 4).
What are parallel and perpendicular lines?
Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals, so their product is −1: slopes 2 and −1/2 are perpendicular.
Forgetting the sign when taking a negative reciprocal. The reciprocal of 3/4 is 4/3, and the negative reciprocal is −4/3.
To write a model from a story, the rate of change becomes the slope and the starting value becomes the y-intercept.
Check any equation by substituting the original points. If both points work, the equation is right.
- Slope = (y₂ − y₁) ÷ (x₂ − x₁) for any two points on the line.
- In y = mx + b, m is the slope and b is the y-intercept.
- Point-slope form needs one point and the slope.
- Parallel lines have equal slopes; perpendicular slopes multiply to −1.
- In a model, the rate of change is the slope and the starting value is the y-intercept.
Writing and Graphing Linear Functions in Three Forms: frequently asked questions
How do you convert standard form to slope-intercept form?
Solve for y. For 4x + 2y = 10, subtract 4x to get 2y = −4x + 10, then divide by 2 to get y = −2x + 5. The slope is −2 and the y-intercept is 5.
When should you use point-slope form?
Use it when you know a point and the slope, or can find the slope from two points. Substitute the numbers directly and then simplify if another form is needed.
What is a negative reciprocal?
Flip the fraction and change the sign. The negative reciprocal of 2/5 is −5/2. Two lines with these slopes are perpendicular because the slopes multiply to −1.
How can you tell from a table that a function is linear?
Equal steps in x must give equal steps in y. For x = 0, 2, 4, 6 and y = 5, 11, 17, 23, y increases by 6 each time, so the rate of change is 3 and the table is linear.
What do slope and intercept mean in a real situation?
The slope is the amount the output changes for each unit of input, such as dollars per month. The y-intercept is the starting value when the input is zero, such as a joining fee.