Linear Functions

Linear functions are the major component of Algebra 1. Master slope and the three equation forms and you'll recognize them everywhere, like in graphs, tables, word problems, and systems.

Click here for some tips on linear equations

Linear function questions come in a few forms: finding slope between two points, converting between equation forms, interpreting slope and y-intercept in context, graphing from an equation, and writing an equation from given information.

• Slope-intercept form: y = mx + b — read slope (m) and y-intercept (b) directly

• Standard form: Ax + By = C — rearrange to slope-intercept to graph

• Point-slope form: y − y₁ = m(x − x₁) — use when you know a point and the slope

• Slope formula: m = (y₂ − y₁) / (x₂ − x₁)

• Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals

✓ Interpreting slope and y-intercept in context is almost always on the exam. The slope is the rate of change ("per unit"). The y-intercept is the starting value ("when x = 0"). Always answer in a full sentence using the units given in the problem.

Worked example

Linear Functions and Slope.

Take a look at this example below (and delete the lines if you want to try it yourself). If you need more help, click the link below the box.

Problem: A car rental company charges a flat fee of $25 plus $0.40 per mile. Write a function for the total cost C after m miles. What is the slope and what does it mean in context?

Things to remember:

1. The y-intercept (25) is the flat fee — what you pay before driving a single mile.

2. The slope (0.40) is the rate of change — cost increases by $0.40 per additional mile.

   
   
   
   

If you found this scratchpad helpful, get our Chrome extension to use it with almost any algebra problem!

Try it yourself! Below is an exam-style problem

Problem: Find the slope of the line passing through (2, 5) and (6, 13). Then write the equation in slope-intercept form.

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Click here to see the answer

STEP-BY-STEP SOLUTION

1. Solution:

   1. Slope: m = (13 − 5) / (6 − 2) = 8 / 4 = 2

   2. Use point-slope with (2, 5): y − 5 = 2(x − 2)

   3. Simplify: y = 2x + 1

For more practice with linear equations, check out these resources:

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:linear-equations-graphs

https://www.ixl.com/math/grade-8/convert-a-linear-equation-in-standard-form-to-slope-intercept-form