Forms of a straight line

A straight line is a graph in which the rate of change between $y$ and $x$ stays constant. In coordinate form, it is usually written as

Here, $x$ is the horizontal coordinate (unitless unless a context gives units) and $y$ is the vertical coordinate (unitless unless a context gives units). The gradient is $m$ (a unitless ratio, or units of $y$ per unit of $x$ in context), while $c$ gives the $y$-intercept value (same units as $y$).

The same line can also be written in general form,

where $a$, $b$ and $d$ are constants (unitless in pure coordinate geometry). Another option is point-gradient form,

where $x_1$ is the horizontal coordinate of a known point and $y_1$ is its vertical coordinate. The guide uses $d$ for the constant term. Many textbooks use $c$ in general form, but don’t confuse it with the $c$ in $y=mx+c$.

All three forms represent the same line. Gradient-intercept form displays the slope and $y$-intercept straight away. Point-gradient form is quickest when a point and a gradient are known, while general form often gives a neat final answer when fractions appear.

Gradient and intercepts

A gradient measures the vertical change divided by the horizontal change along a line. Given two distinct points $(x_1,y_1)$ and $(x_2,y_2)$,

Here, $x_2$ is the horizontal coordinate of the second point and $y_2$ is its vertical coordinate, with $x_2\ne x_1$. Think of gradient as rise over run. When $m>0$, the line rises from left to right; when $m<0$, it falls. If $m=0$, the line is horizontal. A large value of $|m|$ indicates a steep line.

An intercept is a point where a graph meets an axis. For $y=mx+c$, the $y$-intercept is $(0,c)$. To find the $x$-intercept, put $y=0$; if $m\ne0$, the point is $\left(-\frac{c}{m},0\right)$. Watch the wording here: an intercept is a point, not merely a number.

Either two distinct points, or one point together with its gradient, determine a line uniquely. In practical contexts, gradients may describe inclines such as roads, bridges and access ramps. Units then matter because the vertical and horizontal quantities may be measured in metres or as percentages.

Parallel and perpendicular lines

Two non-vertical straight lines are parallel lines when they are distinct and have the same gradient. If $m_1$ is the gradient of the first line and $m_2$ is the gradient of the second line, then

Two non-vertical straight lines are perpendicular lines when they meet at a right angle. Their gradients obey

In other words, one gradient is the negative reciprocal of the other. A steep positive line has a perpendicular that is shallow and negative; that small visual picture is useful to remember.

This topic introduces a major idea in functions: one relationship can be shown geometrically or algebraically. Descartes' coordinate geometry made that connection explicit—a line on a page and an equation can hold the same mathematical knowledge.

What a function is

A relation is a set of ordered pairs connecting elements of one set with elements of another. A function is a relation that assigns every input in its domain to exactly one output. Exactly one—that’s the key condition.

We write

Another notation is

For example, once the domain is understood, $f(x)=2-x$ is shorthand for a rule.

A domain is the set of input values permitted for a function. A codomain is the target set where the outputs are expected to lie, while a range contains the output values the function actually produces. If an IB question doesn’t state the domain, use the largest possible domain on which the function is defined. This is usually $\mathbb{R}$.

To find the largest possible real domain, rule out illegal operations. You cannot divide by zero. In the real numbers, you also cannot take an even root of a negative number or a logarithm of a non-positive number. For instance, $f(x)=\sqrt{2-x}$ has domain $x\le2$ and range $f(x)\ge0$.

Graphs, models and notation

A graph of a function consists of the points $(x,f(x))$ in the coordinate plane as $x$ runs through the domain. For real-valued functions, the vertical line test says that a graph represents a function if every vertical line meets it at no more than one point.

Function notation often gives a clue about what’s being modelled. In a height function $h(t)$, for example, $t$ might represent time. For a velocity function $v(t)$, the output is the velocity at time $t$. In a cost function $C(n)$, the input might be a number of items. A mathematical model uses a function, equation, graph or table to represent a real situation after simplifying assumptions have been made. Natural function models include temperature conversions, currency conversions, projectile height and cost functions.

These assumptions affect the model. We choose which variables to include or ignore, then decide on a sensible domain. A time model may require $t\ge0$, even when its algebraic rule accepts every real number.

Range and the horizontal line test

The graph often makes the range easiest to identify: look at which heights it reaches. The horizontal line test checks whether each horizontal line meets the graph at most once. It can show whether a function is one-to-one and can also help identify the range.

A one-to-one function is a function where each output in the range comes from exactly one input in the domain. Algebraically, $f(a)=f(b)$ implies $a=b$, where $a$ and $b$ belong to the domain. On a graph, a one-to-one function passes the horizontal line test.

Inverse functions as undoing

An inverse function reverses the mapping of a one-to-one function. We write it as $f^{-1}(x)$, with $f^{-1}$ read as inverse of $f$—not reciprocal of $f$. The reciprocal is $\frac{1}{f(x)}$.

When $f$ has an inverse, the domain of $f^{-1}$ is the range of $f$. Its range is the domain of $f$. Solving $f(x)=10$ is equivalent to evaluating $f^{-1}(10)$, provided $10$ lies in the range of $f$.

An inverse swaps the input and output on a graph. A point $(x,y)$ on $y=f(x)$ becomes $(y,x)$ on $y=f^{-1}(x)$. The two graphs are therefore reflections in the line $y=x$.

To find an inverse algebraically, begin with $y=f(x)$. Interchange $x$ and $y$, then solve for $y$. If the original function isn’t one-to-one, restrict its domain before taking the inverse. For example, $f(x)=x^2$ on $\mathbb{R}$ has no inverse function, whereas $f(x)=x^2$ on $x\ge0$ does.

Many mathematicians shaped modern function notation over several centuries, including Descartes, Leibniz and Euler. Its history reminds us that mathematical notation is a language developed by people. It is precise and compact, and it works internationally.

The equation $y=f(x)$

The equation

This form lets you move between a symbolic rule, a table of values, a graph and a real context.

A sketch is a labelled graph showing the general shape of a relationship and its important features. By contrast, a drawn graph is accurate: points are plotted carefully and scales are used properly. In IB wording, sketch doesn’t mean careless. The key features simply matter more than ruler-perfect plotting.

Label every axis and mark important points or asymptotes. When the variables come from a context, the axes should also show their meanings and units. For example, a graph of height against time isn’t complete if its axes are labelled only $x$ and $y$.

Creating graphs from information or context

To create a graph from given information, start with its features: intercepts, turning points, asymptotes, end behaviour, symmetry and any domain restrictions. For a function drawn from a context, choose a sensible window based on that context. Time after an event usually starts at $0$; a population isn’t negative; a cost model may make sense only for whole-number inputs.

Technology forms part of this syllabus. A graphing calculator or graphing package can display functions, including those not named explicitly elsewhere in topic 2. Transfer the graph intelligently. Choose a useful window, label the axes and copy the relevant features instead of reproducing a random screen view.

Sums and differences of functions

Given two functions, their sum function has outputs equal to the sum of their outputs. We write

A difference function is written in a similar way:

Graphing technology is especially useful here. Enter $f(x)$, $g(x)$ and then $f(x)+g(x)$ or $f(x)-g(x)$ to see how the relationship changes. That isn’t just button-pushing; the graph gives another representation of the same mathematical relationship.

There’s a TOK point here. A graph may reveal behaviour hidden by an equation, whereas an equation can prove facts that a graph only suggests. Good mathematics uses both forms because each representation answers different questions well, not because one is always superior.

Key features of graphs

A key feature is a property of a graph that helps describe or solve a problem. For this topic, you should be able to identify maximum and minimum values, intercepts, symmetry, vertices, zeros, roots and vertical or horizontal asymptotes, often with graphing technology.

A maximum value is an output value greater than or equal to nearby or permitted output values. By contrast, a minimum value is an output value less than or equal to nearby or permitted output values. The turning point of a parabola is called a vertex. A zero of a function is an input value for which $f(x)=0$; that same input is also a root of the equation $f(x)=0$.

An asymptote is a line that a graph approaches but does not necessarily meet. A vertical asymptote has equation

A horizontal asymptote has equation

Graphing tools can locate features accurately, but always check whether the result makes sense. The calculator window may hide an asymptote or miss a second intersection. It can also make a steep curve appear vertical.

Intersections of curves and lines

An intersection point lies on two graphs at the same time. For graphs $y=f(x)$ and $y=g(x)$, the intersections solve

Graphing technology can locate intersections directly. Alternatively, graph the difference function

The graphs of $f$ and $g$ intersect exactly when $d(x)=0$.

Graphs are useful in economics and science for this reason. Market equilibrium, production possibilities models, experimental trends and geographical profiles often come down to interpreting key features or intersections. The visual approach doesn’t replace algebra; it offers a different route into the same relationship.

Composite functions

A composite function is formed when one function is applied, followed by another. We write

Order matters. In general,

The algebra may produce different expressions. Sometimes, one of the compositions may not even exist: the output of the first function must lie in the domain of the next function.

For example, let $f(x)=3x-1$ and $g(x)=x^2$. Then $(f\circ g)(x)=3x^2-1$, whereas $(g\circ f)(x)=(3x-1)^2$. The same two functions, applied in a different order, give different results.

Identity function

The identity function returns each input unchanged. It is written

Composing a function with the identity changes nothing:

and

Finding inverse functions and checking them

For a one-to-one function, the inverse reverses what the original function does. Algebraically,

and

Equivalently,

and

These formulas provide a check. If you claim that $g(x)$ is the inverse of $f(x)$, show both $(f\circ g)(x)=x$ and $(g\circ f)(x)=x$ on the relevant domains.

For a composite function, the inverse reverses the order:

This is the socks-and-shoes rule: the inverse undoes the last action first. It also shows how mathematics works as a language—a compact string of notation can describe a whole process.

To find an inverse, write $y=f(x)$, swap $x$ and $y$, then solve for $y$. State the domain and range when relevant. If the function is not one-to-one on its given domain, it has no inverse function unless the domain is restricted.

Equations as intersections and zeros

To solve an equation, find every value of the unknown that makes both sides equal. Graphically, solving

amounts to locating the intersections of the graphs $y=f(x)$ and $y=g(x)$. An analytic solution uses algebra to isolate or determine $x$.

It often helps to move every term to one side. The equation $f(x)=g(x)$ is equivalent to

Solving the equation can therefore be viewed as finding the zeros of a function.

Analytic methods where they fit

Use algebra when the equation’s structure suggests it. For example,

can be handled as a quadratic in $e^x$. Let

Since $e^{2x}=u^2$, the equation becomes

Solve for $u$, then convert back using $u=e^x$. Get into the habit of spotting useful structure, such as powers and repeated expressions. Factorable forms and inverse operations can also point towards an analytic method.

Technology when algebra is not sensible

Some equations have no tidy analytic route at this level. Equations such as

or

may be solved with graphing technology. Plot each side as a separate graph and find their intersections, or graph the difference and locate its zeros. HL students may also be expected to use technology when no appropriate analytic approach is available. At SL, sensible use of graphing tools is already expected.

When using technology, give answers to an appropriate accuracy. Check the viewing window too, since it may hide solutions. If the context restricts the domain, search only within that domain.

Applications and assumptions

Graphing and equation solving arise naturally in real-life models. Examples include radioactive decay, population growth or decay, compound interest, projectile motion and braking distance models. A question might ask when a quantity reaches a threshold or when two quantities are equal. It could also ask when a maximum safe value is exceeded.

For example, suppose $P(t)$ models population after time $t$. Solving

gives the time at which the population reaches level $K$

The algebra is only as reliable as the assumptions behind the model. These may include constant growth rates, ignored external factors, continuous rather than discrete change, and a meaningful domain.

Even and odd functions

An even function gives the same output when the sign of the input is reversed. In algebraic form,

for every $x$ in the domain, where the domain is symmetric about $0$. The graph has reflection symmetry in the $y$-axis.

An odd function gives an output with the opposite sign when the input’s sign is reversed. Algebraically,

for every $x$ in the domain, again with a domain symmetric about $0$. Its graph has rotational symmetry of $180^\circ$ about the origin.

Periodic functions may also be even or odd. Cosine-type symmetry, for instance, is even, while sine-type symmetry is odd. After transformations, don’t decide from the function’s name alone; test $f(-x)$.

Most functions are neither even nor odd. A polynomial containing only even powers of $x$, together with a constant term, is even. If it contains only odd powers of $x$ and has no constant term, it is odd. For an odd function with domain $\mathbb{R}$, $f(0)=0$ because $f(0)=-f(0)$.

Inverses with domain restriction

To find an inverse, use the usual algebraic method: write $y=f(x)$, swap $x$ and $y$, then solve for $y$. At HL, you also need to watch the domain restriction carefully.

For example, $f(x)=x^2$ is not one-to-one on $\mathbb{R}$, so it has no inverse function on that domain. Restricting the domain to $x\ge0$ gives the inverse $f^{-1}(x)=\sqrt{x}$ with domain $x\ge0$. If the domain is restricted to $x\le0$, the inverse is instead $f^{-1}(x)=-\sqrt{x}$. The rule is the same, but a different branch gives a different inverse.

Notation isn’t merely cultural decoration. The domain written beside a function can completely change the meaning of its inverse.

Self-inverse functions

A self-inverse function equals its own inverse. If $f$ is self-inverse, then

and so

The graph is symmetric in the line $y=x$. Taking an inverse reflects a graph in this line, and for a self-inverse function the graph stays unchanged.

Examples include $f(x)=x$, $f(x)=-x$, and $f(x)=\frac{1}{x}$ on domain $x\ne0$. More generally, $f(x)=\frac{k}{x}$ is self-inverse when $k$ is a non-zero real constant, since applying the function twice returns $x$.

The historical development of function notation shows that mathematical symbols are shared conventions. They developed through particular cultures and historical periods. Once agreed, though, they allow mathematicians to communicate structure with remarkable precision.