Functions and graphs

Functions as relationships

A function is a rule that assigns exactly one output to each allowed input. In physics, functions describe how one physical quantity depends on another. Position may depend on time, temperature may depend on location, and force may depend on displacement.

Function notation makes this dependence explicit. If position depends on time, we may write

$x(t)$

This is read as “x as a function of t.” It does not mean $x$ multiplied by $t$. It means the value of position changes according to the input time.

For example,

$x(t) = 3t + 2$

says that when $t = 0$, $x = 2$, and for every increase of 1 second, $x$ increases by 3 meters if the units are meters and seconds.

Independent and dependent variables

The input is often called the independent variable. The output is the dependent variable. In $x(t)$, time is the independent variable and position is the dependent variable. This language helps you ask: What am I choosing or measuring first? What responds?

In many physics graphs, time is placed on the horizontal axis because motion is often studied as it evolves. Position, velocity, energy, or temperature may be placed on the vertical axis.

Graphs as visual equations

A graph is a picture of a relationship. The horizontal axis shows input values, and the vertical axis shows output values. A point on the graph represents a matched pair.

For the linear function

$y = mx + b$

$m$ is the slope and $b$ is the vertical intercept. The slope measures how much $y$ changes when $x$ changes:

m = rac{Delta y}{Delta x}

In physics, slope often has meaning. On a position-time graph, slope is velocity. On a velocity-time graph, slope is acceleration. On a force-displacement graph, area can represent work.

Interpreting slope physically

Suppose a position-time graph is a straight line with slope $4 m/s$. That means the object moves 4 meters each second in the positive direction. A steeper line means a greater speed. A horizontal line means position is not changing, so velocity is zero.

If the graph curves, the slope changes from point to point. This means the related physical quantity is changing. A curved position-time graph indicates changing velocity, which means acceleration.

Intercepts and meaning

Intercepts are also useful. The vertical intercept tells the output when the input is zero. In $x(t) = 3t + 2$, the intercept 2 means the initial position is 2 meters. The horizontal intercept tells where the output becomes zero, if that value exists.

In physics, always interpret intercepts with units and context. An intercept is not just a number on a graph. It may represent initial position, starting temperature, initial energy, or another meaningful quantity.

Linear and nonlinear functions

Many first physics models are linear because linear relationships are easy to analyze. Hooke's law for a spring is often written

$F = kx$

This says force is proportional to displacement. Double the stretch, double the force.

But many physical relationships are nonlinear. Kinetic energy depends on speed squared:

K = rac{1}{2}mv^2

Doubling speed quadruples kinetic energy. Graphs help reveal these differences.

Domain and range

The domain is the set of allowed inputs. The range is the set of possible outputs. In physics, domain and range are often limited by reality. Time may not be negative for a particular experiment. Speed may be nonnegative. A square root formula may require the expression inside the root to be nonnegative.

The big idea

Functions and graphs turn physical relationships into visual and symbolic tools. A function tells how one quantity depends on another. A graph shows that dependence in a form your eyes can interpret. Learning to read slope, intercepts, curvature, domain, and range builds the foundation for calculus and advanced physics.