Functions & Graphs

Draw and interpret linear, quadratic, hyperbolic and exponential graphs

What is a Function?

A function is a rule that assigns exactly one output to each input. We write y = f(x). The input is x (independent variable) and the output is y (dependent variable). The graph of a function shows all the (x, y) points that satisfy the rule.

Linear Functions

A linear function has the form y = mx + c. The graph is a straight line. 'm' is the gradient (slope) — it shows how steep the line is. 'c' is the y-intercept — where the line crosses the y-axis.

Example

Sketching y = 2x − 3

Gradient m = 2 (line goes up, rises 2 for every 1 across). y-intercept c = −3 (line crosses y-axis at (0, −3)). Another point: when x = 2, y = 2(2) − 3 = 1, so (2, 1). Plot (0, −3) and (2, 1) and draw a straight line through them.

Quadratic Functions (Parabolas)

A quadratic function has the form y = ax² + bx + c. The graph is a parabola. If a > 0, it opens upward (smile). If a < 0, it opens downward (frown). The turning point is at x = −b/(2a).

Example

Hyperbola and Exponential

Hyperbola: y = a/x. Two curves, one in quadrants I and III (if a > 0). Asymptotes at x = 0 and y = 0. Exponential: y = a.b^x. Rapid growth (b > 1) or decay (0 < b < 1). Always passes through (0, a). Asymptote at y = 0.

Note

Remember

Know the shape of each function type: linear (line), quadratic (parabola), hyperbola (two curves), exponential (rapid increase/decrease). Domain and range describe the set of all possible x-values and y-values.

Key Vocabulary

FunctionA rule assigning exactly one output to each input

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

y-interceptWhere the graph crosses the y-axis

ParabolaThe U-shaped graph of a quadratic function

AsymptoteA line that a graph approaches but never touches

DomainThe set of all possible input (x) values

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