KAHIBARO
Table of Contents
Seeing Physics Through Functions
Physics describes how one quantity depends on another. A function is the mathematical tool that expresses this dependence. If the position of a car changes with time, or the temperature of a metal changes as it is heated, we use a function to show the rule connecting input and output.
A function takes an input value and gives exactly one output value. We often write this as
$$
y = f(x)
$$
This means that the value of $y$ is determined by the value of $x$ through the rule called $f$.
In physics, the input is often time, position, angle, or some other measurable quantity. The output may be speed, force, energy, pressure, or many other physical quantities.
For example, if the position of an object depends on time, we may write
$$
x(t) = 3t + 2
$$
This says that position $x$ is a function of time $t$.
A function assigns one output to each allowed input.
In physics notation, $x(t)$ means "the quantity $x$ depends on $t$". It does not mean multiplication.
Variables, Constants, and Parameters
A variable is a quantity that can change. In $x(t)$, the time $t$ is the independent variable, and the position $x$ is the dependent variable.
A constant is a quantity that stays fixed in a given problem. In the expression
$$
x(t) = 3t + 2
$$
the numbers $3$ and $2$ are constants.
Sometimes a formula contains parameters. These are fixed for one situation, but could change in another. For example,
$$
y = mx + b
$$
has parameters $m$ and $b$. In one graph they may have certain values, and in another graph different ones.
Domain and Range
The domain of a function is the set of input values that are allowed. The range is the set of output values the function can produce.
For pure mathematics, a function might allow many values. In physics, the physical meaning often restricts the domain. Time may start at $t = 0$. A mass cannot be negative. A distance may only make sense for certain positions.
For example, if a ball is dropped at $t = 0$, then a position function might only be meaningful for
$$
t \geq 0
$$
If the graph describes a temperature in kelvin, then the output cannot be negative.
In physics, the mathematically possible values are not always physically possible values.
Always check the physical meaning of both the input and the output.
Common Ways to Write a Function
A function can be represented in several useful ways. In physics, it is important to move comfortably between them.
| Form | Example | Meaning |
|---|---|---|
| Words | Position increases steadily with time | Verbal description |
| Equation | $x(t) = 3t + 2$ | Algebraic rule |
| Table | Values of $t$ and $x$ listed | Numerical data |
| Graph | Curve on axes | Visual relationship |
A table for $x(t) = 3t + 2$ might look like this:
| $t$ | $x(t)$ |
|---|---|
| 0 | 2 |
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
The graph gives a quick picture of how the output changes as the input changes.
Reading Graph Axes
A graph shows how one quantity varies with another. The horizontal axis usually shows the independent variable, and the vertical axis shows the dependent variable.
If position depends on time, then time is usually placed on the horizontal axis and position on the vertical axis.
It is very important to label axes clearly with both quantity and unit. For example,
$$
t \text{ in s}, \qquad x \text{ in m}
$$
Without units, a graph in physics is incomplete.
A physics graph should identify both the variable and its unit on each axis.
For example, write $t \, (\text{s})$ and $x \, (\text{m})$, not just $t$ and $x$.
Interpreting the Shape of a Graph
The shape of a graph carries physical meaning. Even before learning derivatives, you can understand many basic features.
If the graph rises as you move to the right, the output increases as the input increases. If it falls, the output decreases. If it is horizontal, the output stays constant.
A straight line means the quantity changes at a constant rate. A curved graph means the rate of change itself is changing.
Here are some common graph shapes:
| Graph type | Example form | Basic meaning |
|---|---|---|
| Horizontal line | $y = c$ | Constant quantity |
| Straight line | $y = mx + b$ | Constant rate of change |
| Parabola | $y = ax^2 + bx + c$ | Rate changes steadily |
| Sinusoidal curve | $y = A\sin(\omega t)$ | Repeating behavior |
| Exponential curve | $y = Ae^{kt}$ | Growth or decay |
These shapes appear often in physics. At this stage, the key goal is to recognize and interpret them qualitatively.
Linear Functions
A linear function has the form
$$
y = mx + b
$$
The constant $m$ is the slope, and $b$ is the vertical intercept.
The slope tells how much $y$ changes when $x$ changes by one unit:
$$
m = \frac{\Delta y}{\Delta x}
$$
If $m > 0$, the graph rises. If $m < 0$, it falls. If $m = 0$, it is horizontal.
In physics, many simple relationships are linear over some range. For example, a steadily moving object has a position that changes linearly with time.
For a straight line,
$$
m = \frac{\Delta y}{\Delta x}
$$
This is the slope, and it measures the rate of change of the graph.
Nonlinear Functions
Many physical relationships are not linear. A falling object, an oscillating spring, and radioactive decay all produce curved graphs.
A nonlinear function may still be simple and predictable. For example,
$$
y = x^2
$$
is not a straight line. As $x$ increases, $y$ increases faster and faster.
Another important example is the inverse relationship:
$$
y = \frac{1}{x}
$$
Here, increasing $x$ makes $y$ smaller.
In physics, nonlinear behavior is common, so learning to read curved graphs is essential.
Intercepts and Special Points
A graph often has important points that help us understand the function.
The vertical intercept is where the graph crosses the vertical axis. It is found by setting the horizontal variable equal to zero. In
$$
y = 2x + 5
$$
the vertical intercept is $5$, because when $x = 0$, $y = 5$.
The horizontal intercept is where the graph crosses the horizontal axis. It is found by setting the vertical variable equal to zero. In the same example,
$$
0 = 2x + 5 \quad \Rightarrow \quad x = -\frac{5}{2}
$$
In physics, intercepts can have direct meaning. A position at $t = 0$ is an initial position. A velocity reaching zero can mark a turning point.
Maximum and Minimum Values
Some functions have highest or lowest points. These are called maxima and minima.
A ball thrown upward reaches a maximum height before falling down. A potential energy curve may have a minimum at a stable equilibrium position.
For a simple parabola such as
$$
y = -x^2 + 4
$$
the graph has a maximum at the top.
Recognizing such points helps us identify important events in physical systems.
Graphing From a Table of Data
In experiments, functions often begin as data rather than formulas. You may measure time and position, or voltage and current, then plot the points on a graph.
A graph from data helps reveal patterns. If the points lie nearly on a straight line, the relationship may be approximately linear. If they form a curve, another type of function may describe the data better.
When plotting data, choose scales that make the pattern clear. Mark points carefully and label axes with units.
Graphing From an Equation
If you are given a formula, you can graph it by computing several values and plotting them. For example, for
$$
y = x^2
$$
you can make a table:
| $x$ | $y$ |
|---|---|
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
Then plot these points and connect them smoothly.
This process is especially useful when first learning unfamiliar functions.
Comparing Different Graphs
Physics often compares several quantities that depend on the same variable. For example, different objects may have different position-time graphs. Comparing them can show which object starts farther away, which changes faster, or which reaches a certain value first.
A steeper graph usually means a greater rate of change. A higher intercept means a larger initial value. A curved graph means behavior is changing in time or space.
Even without advanced calculus, careful graph reading gives a great deal of physical insight.
Piecewise Functions
Some physical situations follow different rules in different regions. Then we use a piecewise function.
For example,
$$
f(t) =
\begin{cases}
0, & t < 2 \\
3, & t \geq 2
\end{cases}
$$
This function stays at $0$ until $t = 2$, then jumps to $3$.
Piecewise functions are useful for switches turning on, forces applied only during part of a motion, or motion that changes rule after a collision.
Visual Example of a Graph
This graph shows a linear relation between position and time. The line rises steadily, so position increases at a constant rate.
Why Graphs Matter in Physics
Graphs help us see relationships quickly. They reveal patterns that may be hard to notice in equations alone. A graph can show whether a quantity is constant, increasing, decreasing, repeating, or changing more rapidly over time.
They also help connect theory and experiment. A theoretical formula predicts a graph, and experimental data can be plotted to test whether reality matches the prediction.
As physics becomes more advanced, graphs will help you understand motion, forces, energy, waves, electric circuits, and many other topics.
A graph is not just a picture. In physics, it is a compact description of how one measurable quantity depends on another.
Final Perspective
Functions and graphs are the language of change and relationship in physics. A function tells the rule, and a graph shows the rule visually. By learning to recognize variables, slopes, intercepts, shapes, and physical restrictions, you build a foundation for nearly every topic that follows.