Algebra 1 Linear Functions- Study Guide
What Linear Functions Actually Are
A linear function is any equation that graphs as a straight line. That's it. No curves, no loops, just a flat line going left to right at some angle.
The standard form is:
f(x) = mx + b
Or in plain math terms: y = mx + b
Every linear function has exactly two parts that control everything about the line: m (the slope) and b (the y-intercept). Get these two numbers right and you can graph any line, write any equation, and solve any problem they throw at you.
The Two Numbers That Control Everything
Slope (m)
Slope tells you how steep the line is and which direction it goes. You calculate it as:
Slope = rise / run = (y₂ - y₁) / (x₂ - x₁)
Think of it as "how much y changes when x changes by 1."
Here's how slope values translate:
- Positive slope: line goes upward left to right 📈
- Negative slope: line goes downward left to right 📉
- Slope of 0: horizontal line (flat)
- Undefined slope: vertical line (|)
A slope of 3 means: for every 1 unit you move right, the line goes up 3 units.
Y-Intercept (b)
The y-intercept is where the line crosses the y-axis. This happens when x = 0. So plug in x = 0 and whatever y equals is your b value.
In y = 2x + 5, the line crosses the y-axis at (0, 5).
The Three Forms You Need to Know
Linear equations show up in three main formats. Each one is useful in different situations.
Slope-Intercept Form: y = mx + b
This is the most common form and the one you should default to. You can read the slope and y-intercept directly without any work.
Example: y = -3x + 7 → slope is -3, y-intercept is 7
Point-Slope Form: y - y₁ = m(x - x₁)
Use this when you know one point on the line and the slope. That's all you need.
Example: A line with slope 4 passing through (2, 5)
y - 5 = 4(x - 2)
Standard Form: Ax + By = C
The A, B, and C are integers, and A should be positive. This form is useful for finding intercepts and working with certain algebraic operations.
Example: 2x + 3y = 12
| Form | Structure | Best Used When |
|---|---|---|
| Slope-Intercept | y = mx + b | You know slope and y-intercept, or need to graph quickly |
| Point-Slope | y - y₁ = m(x - x₁) | You know one point and the slope |
| Standard | Ax + By = C | Finding intercepts, working with integer coefficients |
How to Graph Linear Functions
Graphing is straightforward once you know the two key numbers:
- Plot the y-intercept (0, b) on the y-axis
- Use the slope to find another point: rise/run from your starting point
- Connect the dots with a straight line
- Extend in both directions and add arrows at the ends
Quick example: Graph y = 2x + 3
- Plot (0, 3) — the y-intercept
- Slope is 2/1: go up 2, right 1 → land on (1, 5)
- Draw the line through these points
That's all. Takes about 30 seconds once you get the hang of it.
Writing Equations From Different Information
From Two Points
- Calculate slope using the formula: m = (y₂ - y₁) / (x₂ - x₁)
- Plug one point and the slope into y - y₁ = m(x - x₁)
- Simplify to slope-intercept form if needed
From a Graph
- Find the y-intercept on the y-axis
- Pick two clear points and calculate slope
- Write y = mx + b with your values
From the Slope and One Point
Plug directly into point-slope form, then solve for y to get slope-intercept form.
Parallel vs. Perpendicular Lines
This shows up constantly on tests.
- Parallel lines have the same slope but different y-intercepts
- Perpendicular lines have slopes that are negative reciprocals: if one has slope 3, the other has slope -1/3
Check: 3 × (-1/3) = -1. The product of perpendicular slopes is always -1.
Common Mistakes That Cost Points
- Mixing up signs when calculating slope — double-check (y₂ - y₁) / (x₂ - x₁)
- Forgetting the y-intercept when writing equations from slope and a point
- Confusing slope of 0 (horizontal line) with undefined slope (vertical line)
- Not converting between forms when the problem requires a specific format
Practice Strategy That Actually Works
Don't just read examples. Work problems. Start with graphing from y = mx + b, then work backwards: given a graph, write the equation.
When you get a problem wrong, figure out exactly where you made the error. Is it identifying the slope? Reading the graph wrong? Sign errors? Isolating the variable wrong? Know your specific weakness and drill it.
You need maybe 20-30 practice problems covering all the variations before this becomes automatic.