Patterns & Functions

Linear functions and their graphs

Linear Functions and Their Graphs

A linear function has the form y = mx + c, where m is the gradient (slope) and c is the y-intercept (where the line crosses the y-axis). The graph is always a straight line.

Example

Finding the Equation

A line passes through (0, 3) and (2, 7). Gradient m = (7−3)/(2−0) = 4/2 = 2 y-intercept c = 3 (where x = 0) Equation: y = 2x + 3 Check: at x = 2, y = 2(2) + 3 = 7 ✓

Interpreting Gradient and y-Intercept

• Positive gradient: line goes uphill (left to right) • Negative gradient: line goes downhill • Steep line: large |m| • y-intercept: starting value when x = 0 • x-intercept: let y = 0 and solve for x

Note

Remember

y = mx + c is the standard form. Gradient = rise/run = (y₂−y₁)/(x₂−x₁). Parallel lines have equal gradients. A horizontal line has gradient 0. A vertical line has undefined gradient.

Key Vocabulary

Linear functionA function whose graph is a straight line (y = mx + c)

GradientThe slope of a line (m in y = mx + c)

y-interceptThe point where a graph crosses the y-axis

x-interceptThe point where a graph crosses the x-axis

SASL Avatar

Loading avatar...

1 / 4

Linear function

Speed: