Linear Functions Formula

The Formula

y = mx + b (slope m, y-intercept b)

When to use: Every step right changes y by the same amount—like climbing stairs at a constant pace.

Quick Example

y = 2x + 3: at x=0, y=3; at x=2, y=7. The slope is always 2.

Notation

m is slope, b is y-intercept. Slope-intercept form: y = mx + b. Point-slope form: y - y_1 = m(x - x_1). Standard form: Ax + By = C.

What This Formula Means

A function whose graph is a straight line, characterized by a constant rate of change between any two points.

Every step right changes y by the same amount—like climbing stairs at a constant pace.

Formal View

A function f: \mathbb{R} \to \mathbb{R} is linear if \exists\, m, b \in \mathbb{R} such that f(x) = mx + b for all x. Equivalently, f is linear iff \frac{f(x_2) - f(x_1)}{x_2 - x_1} = m for all x_1 \neq x_2.

Worked Examples

Example 1

easy
Write the equation of a line with slope 2 and y-intercept -3.

Solution

  1. 1
    The slope-intercept form is y = mx + b.
  2. 2
    Substitute m = 2 and b = -3.
  3. 3
    The equation is y = 2x - 3.

Answer

y = 2x - 3
In slope-intercept form y = mx + b, the parameter m is the slope (rate of change) and b is the y-intercept (the value of y when x = 0).

Example 2

medium
Find the equation of the line through (1, 3) and (4, 12).

Example 3

medium
Write the equation of a line with slope 3 that passes through (1, 7).

Common Mistakes

  • Confusing slope with y-intercept
  • Graphing from wrong point

Why This Formula Matters

Models constant-rate processes: cost, distance, temperature, etc.

Frequently Asked Questions

What is the Linear Functions formula?

A function whose graph is a straight line, characterized by a constant rate of change between any two points.

How do you use the Linear Functions formula?

Every step right changes y by the same amount—like climbing stairs at a constant pace.

What do the symbols mean in the Linear Functions formula?

m is slope, b is y-intercept. Slope-intercept form: y = mx + b. Point-slope form: y - y_1 = m(x - x_1). Standard form: Ax + By = C.

Why is the Linear Functions formula important in Math?

Models constant-rate processes: cost, distance, temperature, etc.

What do students get wrong about Linear Functions?

The y-intercept is where the line crosses the y-axis (when x = 0).

What should I learn before the Linear Functions formula?

Before studying the Linear Functions formula, you should understand: slope, equations, coordinate plane.

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