doPhysics.in
All chapters
doPhysics · Mathematical Tools · Toolbox

Functions and Graphs

Start reading0%done
Why this chapter
Why this chapter exists

You may go on with the physics now. When a lesson needs this tool, it will tell you and bring you here.

Physics speaks in graphs. A Metro train's journey is one v-t line. A gas squeezed at one temperature traces a hyperbola, and Planck's constant is the slope of a straight line. The same few shapes return in every chapter: the line, the parabola, the hyperbola and the circle. This chapter teaches each shape as the physics uses it, and teaches the five things any graph can tell you. Each lesson ends with a worked example from a physics chapter of this book.

You will learn
By the end of this chapter you will
  1. Treat a formula as a function, and say which quantity goes on which axis and over what domain.
  2. Read the slope and the intercepts of a straight line as physical quantities with units.
  3. Use the square law and the inverse-square law by ratios, and find the vertex of a parabola.
  4. Read the height, slope, area, intercepts and turning points of any graph in the book.
  5. Find distances, the point that divides a line, and the equation and tangent of a circle.
Before you begin
Before you begin

You should be able to substitute numbers into a formula and plot points on squared paper. Squares, square roots and simple fractions are enough for most of the chapter. The lesson on reading graphs uses the ideas of a slope and an area; the chapters Differentiation and Integration explain them fully.

Already know this?

Lessons

1
A Function and Its Graph
A function gives exactly one output for each input. The independent and the dependent variable, the domain, which quantity goes on which axis, and how \(f(x - a)\) moves a graph without changing its shape.
A function and its graphMoving a graph without changing its shape
13 min0%
→
2
The Straight Line: Slope and Intercept
The straight line \(y = mx + c\). What the slope and the intercept mean as physics, with their units, where the line meets the horizontal axis, why a line is not always a proportion, and the line through a point with a given slope.
The straight line slope and interceptSlope and intercept are physical quantitiesThe line through a point with a given slope
15 min0%
→
3
The Parabola and the Square Law
In the square law \(y = kx^2\), twice the input gives four times the output. The distances of a falling stone, the vertex of \(y = ax^2 + bx + c\) found by completing the square, and the top of a thrown ball's flight.
The parabola and the square lawThe vertex of a parabola
14 min0%
→
4
Inverse and Inverse-Square Laws
In \(y = k/x\) the product \(xy\) never changes, and the graph is a hyperbola that never reaches its axes. In \(y = k/x^2\) twice the distance gives a quarter of the effect. Boyle's law, gravity at the Moon, and how to turn a hyperbola into a straight line.
Inverse and inverse square lawsThe inverse square law falls faster
14 min0%
→
5
Reading a Graph: Slope, Area and Intercept
Every graph in physics can be read five ways: its height, its slope, the area under it, its intercepts and asymptotes, and its turning points. The units say what each reading means. A Metro train's v-t graph and the work under a p-V path.
Reading slope area and intercept of a graphIntercepts asymptotes and turning points
15 min0%
→
6
Coordinate Geometry for Physics
Points as pairs of numbers. The distance between two points in a plane and in space, the distance from an axis, the point that divides a line in a given ratio and the centre of mass, the circle, perpendicular lines and the tangent to a circle in point-slope form.
Coordinate geometry for physicsThe section formula and the centre of massThe circle and its tangent
14 min0%
→

Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

0%
The Workbook
Questions to solve on paper, marked by you.
Open →

Glossary

The key words of this chapter, with the meaning this chapter gives them.

asymptote
A line that a graph approaches more and more closely but never reaches. Examples are the axes for \(y = k/x\) and the terminal velocity on a v-t graph.
circle
The set of all points at one distance \(r\) from a centre \((a, b)\), with the equation \((x - a)^2 + (y - b)^2 = r^2\).
dependent variable
The output of a function, the quantity that is measured or that follows from the input. It is plotted along the vertical axis.
domain
The set of inputs for which a function is defined and, in physics, for which the formula describes what really happens.
function
A rule that gives exactly one output for each allowed input, written \(y = f(x)\).
hyperbola
The graph of an inverse law \(y = k/x\). Every point of it makes a rectangle of the same area \(k\) with the axes.
independent variable
The input of a function, the quantity that is chosen or set, such as time. It is plotted along the horizontal axis.
intercept
The value at which a graph meets an axis. For \(y = mx + c\), the intercept on the \(y\) axis is \(c\) and on the \(x\) axis it is \(-c/m\).
inverse-square law
A law of the form \(y = k/x^2\), in which twice the distance gives a quarter of the effect. Gravity and the intensity of a point source follow it.
parabola
The graph of a quadratic, \(y = ax^2 + bx + c\). It opens upward for \(a > 0\) and downward for \(a < 0\).
slope
The rise divided by the run between two points of a straight line, \(m = \Delta y/\Delta x\); on a curve, the slope of the tangent. Its unit is the unit of \(y\) per unit of \(x\).
turning point
A point of a graph where the slope is zero, so that the graph is momentarily flat: a greatest or a least value.
vertex
The turning point of a parabola, at \(x = -b/2a\). It is the lowest point when the parabola opens upward and the highest when it opens downward.