Maths AA terms in Functions

The size of a real number without its sign: $\mathrm{\mid }x\mathrm{\mid }=x$ when $x\ge 0$ and $\mathrm{\mid }x\mathrm{\mid }=-x$ when $x<0$. $\mathrm{\mid }a-b\mathrm{\mid }$ is the distance between $a$ and $b$ on the number line.

A solution found by algebraic manipulation, which gives the exact value, such as $x=\ln 3$ or $x=\frac{1+\sqrt{5}}{2}$, rather than a decimal approximation.

A line that a graph approaches more and more closely, without reaching it, as $x$ or $y$ grows without bound, such as $x=0$ and $y=0$ for $y=\frac{1}{x}$.

The vertical line through the vertex about which a parabola is symmetric. For $y=a{x}^2+bx+c$ it is the line $x=-\frac{b}{2a}$.

Rewriting a quadratic $a{x}^2+bx+c$ in the form $a(x-h{)}^2+k$. The result shows the vertex $(h,k)$ and can be used to solve the equation $a{x}^2+bx+c=0$.

The function $f\circ g$ formed by applying $g$ first and then applying $f$ to the result, so that $(f\circ g)(x)=f(g(x))$.

Compound interest in the limit where the number of compounding periods per year increases without bound. A principal $P$ at nominal annual rate $r$ grows to $A=P{e}^{rt}$ after $t$ years.

A value of $x$ at which an expression in an inequality is zero or undefined. Critical values split the number line into intervals on which the sign of the expression does not change; a sign diagram records that sign on each interval.

The quantity $\mathrm{\Delta }=b^2-4ac$ for the quadratic $a{x}^2+bx+c$. The equation $a{x}^2+bx+c=0$ has two distinct real roots if $\mathrm{\Delta }>0$, two equal real roots if $\mathrm{\Delta }=0$ and no real roots if $\mathrm{\Delta }<0$.

The set of all input values for which a function is defined, or for which it is used in a context.

How the values of a function behave as $x\to \mathrm{\infty }$ and as $x\to -\mathrm{\infty }$. For a polynomial it depends only on the degree and the sign of the leading coefficient.

An equation that becomes a quadratic equation after a substitution such as $u=x^2$, $u=e^x$ or $u=\sin x$; for example, $x^4-5x^2+4=0$ becomes $u^2-5u+4=0$.

A function for which $f(-x)=f(x)$ for every $x$ in its domain. Its graph is symmetric in the $y$-axis, for example $y=x^2$ and $y=\cos x$.

Decrease in which a quantity is multiplied by the same factor between 0 and 1 over each equal time interval, as in $N=N_0{e}^{kt}$ with $k<0$. The time taken to halve, $\frac{\ln 2}{\mathrm{\mid }k\mathrm{\mid }}$, is the half-life.

A function of the form $f(x)=a^x$, where the base $a$ is a positive constant and $a\mathrm{\ne }1$, so that the variable is in the exponent. The natural exponential function is $f(x)=e^x$.

Growth in which a quantity is multiplied by the same factor greater than 1 over each equal time interval, as in $N=N_0{e}^{kt}$ with $k>0$. The time taken to double, $\frac{\ln 2}{k}$, is the doubling time.

For a polynomial $P(x)$, $(x-a)$ is a factor of $P(x)$ if and only if $P(a)=0$. More generally, $(ax-b)$ is a factor if and only if $P\left(\frac{b}{a}\right)=0$.

The form $f(x)=a(x-p)(x-q)$ of a quadratic function, in which $p$ and $q$ are its zeros, so its graph meets the $x$-axis at $(p,0)$ and $(q,0)$.

A relation that assigns to each input in its domain exactly one output. It is written $f(x)$ or $f:x\mapsto \dots$, where $x$ is the input.

The greatest value a function takes on its whole domain. It may occur at a local maximum or at an endpoint of the domain, and some functions have none.

The least value a function takes on its whole domain. It may occur at a local minimum or at an endpoint of the domain, and some functions have none.

The steepness of a straight line: the change in $y$ divided by the change in $x$ between any two points on it, $m=\frac{y_2-y_1}{x_2-x_1}$. For a curve, the gradient at a point is the gradient of the tangent there.

A horizontal line $y=k$ that a graph approaches as $x\to \mathrm{\infty }$ or $x\to -\mathrm{\infty }$, so that $f(x)$ gets as close to $k$ as you like.

The graph of a function is one-to-one, and so the function has an inverse, when every horizontal line meets it at most once.

The curve formed by the graph of $y=\frac{k}{x}$, or of any function $\frac{ax+b}{cx+d}$ that is not constant: two separate branches, each approaching the same pair of perpendicular asymptotes.

The function $I(x)=x$, which maps every input to itself. Composing a function with its inverse, in either order, gives the identity function.

The point that a given point is moved to by a transformation. Under $y=f(x-a)+b$, the image of $(x,y)$ is $(x+a,y+b)$.

In a composite function $f(g(x))$, the function $g$ that is applied to $x$ first. The function $f$, applied to the output of $g$, is the outer function.

A way of writing a set of real numbers between two endpoints. Square brackets include an endpoint and round brackets exclude it: $[2,5)$ means $2\le x<5$.

A point that is mapped to itself by a transformation, so it lies on both the original graph and its image.

The function $f^{-1}$ that undoes $f$: if $f(a)=b$ then $f^{-1}(b)=a$. It exists only when $f$ is one-to-one, and its graph is the reflection of the graph of $f$ in $y=x$.

A relationship between two positive quantities in which their product is constant, $xy=k$, so that $y=\frac{k}{x}$. Doubling one quantity halves the other.

A model of the form $y=mx+c$, in which one quantity changes at a constant rate $m$ with respect to another, starting from the initial value $c$.

A point on a graph that is higher than all nearby points, where the graph changes from increasing to decreasing.

A point on a graph that is lower than all nearby points, where the graph changes from decreasing to increasing.

A function of the form $f(x)={\log }_{a}x$, $x>0$, where $a>0$ and $a\mathrm{\ne }1$. It is the inverse of the exponential function $a^x$; with base $e$ it is $\ln x$.

A function in which at least one output comes from two or more different inputs, such as $f(x)=x^2$, where $f(-3)=f(3)=9$. It has no inverse function on that domain.

A function or equation that describes a real situation, with variables standing for real quantities. Its domain is limited to values that make sense in the context.

The number of times a factor $(x-a)$ appears in the fully factorised form of a polynomial. A root of odd multiplicity is a point where the graph crosses the $x$-axis; a root of even multiplicity is a point where it touches the axis and turns.

The largest set of real numbers for which a formula gives a real output: values that would mean dividing by zero, taking the square root of a negative number or the logarithm of a non-positive number are left out.

A solution found approximately, for example from a graph or by a GDC, given as a decimal to a stated accuracy. It is used when no algebraic method can isolate the variable.

A straight line $y=mx+c$ with $m\mathrm{\ne }0$ that a graph approaches as $x\to \pm \mathrm{\infty }$. A rational function whose numerator has degree exactly one more than its denominator has one, found by polynomial division.

A function for which $f(-x)=-f(x)$ for every $x$ in its domain. Its graph has rotational symmetry of order 2 about the origin, for example $y=x^3$ and $y=\sin x$.

A function in which different inputs always give different outputs, so each output comes from exactly one input. Its graph passes the horizontal line test.

The U-shaped curve that is the graph of a quadratic function. It opens upwards when $a>0$, downwards when $a<0$, and is symmetric about a vertical line through its vertex.

Lines in the same plane that never meet. Two non-vertical lines are parallel when they have equal gradients, $m_1=m_2$, and different $y$-intercepts.

The simplest function of a family, such as $y=x^2$ or $y=\frac{1}{x}$, from which every other member can be obtained by transformations.

Meeting at a right angle. Two lines with gradients $m_1$ and $m_2$ are perpendicular when $m_1\times m_2=-1$, or when one is horizontal and the other vertical.

The line through the midpoint of a line segment at right angles to it. Every point on it is the same distance from the two ends of the segment.

An expression of the form $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$, where $n\in \mathbb{N}$ and $a_n\mathrm{\ne }0$. The degree is $n$, the highest power of $x$; the leading coefficient is $a_n$, the coefficient of that power.

Dividing a polynomial $P(x)$ by a polynomial $D(x)$ to write $P(x)=D(x)Q(x)+R(x)$, where $Q(x)$ is the quotient and the remainder $R(x)$ has a lower degree than $D(x)$.

An equation that can be written as $a{x}^2+bx+c=0$ with $a\mathrm{\ne }0$. It has at most two real solutions, called its roots.

The formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, which gives the solutions of the quadratic equation $a{x}^2+bx+c=0$, where $a\mathrm{\ne }0$.

A function of the form $f(x)=a{x}^2+bx+c$, where $a$, $b$ and $c$ are constants and $a\mathrm{\ne }0$. Its graph is a parabola.

An inequality such as $a{x}^2+bx+c>0$ or $a{x}^2+bx+c\le 0$, where $a\mathrm{\ne }0$. Its solution set is usually an interval or a pair of intervals.

The set of all output values that a function actually takes when $x$ runs through the whole domain.

A function that can be written as a quotient of two polynomials, $f(x)=\frac{p(x)}{q(x)}$, where $q(x)$ is not the zero polynomial. It is undefined wherever $q(x)=0$.

The function $f(x)=\frac{1}{x}$, $x\mathrm{\ne }0$, which maps each non-zero number to its reciprocal. Its graph is a hyperbola with asymptotes $x=0$ and $y=0$.

A transformation that flips a graph in a mirror line. $y=-f(x)$ is the reflection of $y=f(x)$ in the $x$-axis, and $y=f(-x)$ is its reflection in the $y$-axis.

When a polynomial $P(x)$ is divided by $(x-a)$, the remainder is $P(a)$. When it is divided by $(ax-b)$, the remainder is $P\left(\frac{b}{a}\right)$.

A root that occurs twice, as in $a(x-h{)}^2=0$, which happens exactly when $\mathrm{\Delta }=0$. The graph touches the $x$-axis at $x=h$ without crossing it.

A domain chosen as part of the largest possible one, usually so that a many-to-one function becomes one-to-one and has an inverse, for example $f(x)=x^2$, $x\ge 0$.

A function that is its own inverse, so $f^{-1}(x)=f(x)$ and $f(f(x))=x$ for every $x$ in its domain. Its graph is symmetric in the line $y=x$.

To draw a graph by hand that shows its general shape and key features (intercepts, maximum and minimum points, asymptotes, endpoints) labelled with their values, without plotting it to scale.

A transformation that multiplies the distance of every point from an axis by a scale factor. $y=pf(x)$ is a vertical stretch with scale factor $p$; $y=f(qx)$ is a horizontal stretch with scale factor $\frac{1}{q}$.

An inequality using $<$ or $>$, so equality is not allowed. A non-strict inequality uses $\le$ or $\ge$ and includes the values where both sides are equal.

A short method for dividing a polynomial by a linear factor $(x-a)$ that works with the coefficients only: bring down, multiply by $a$, add, and repeat.

A change that moves, flips or resizes a graph by applying a rule to every point on it, such as a translation, a reflection or a stretch.

A transformation that slides every point of a graph the same distance in the same direction, without changing its shape. A translation by the vector $\left(\genfrac{}{}{0px}{}{a}{b}\right)$ maps $y=f(x)$ to $y=f(x-a)+b$.

The point where a parabola turns, lying on its axis of symmetry. It is the minimum point of the graph when $a>0$ and the maximum point when $a<0$.

The form $f(x)=a(x-h{)}^2+k$ of a quadratic function, in which $(h,k)$ is the vertex of its graph and $x=h$ is the axis of symmetry.

A vertical line $x=k$ that a graph approaches as $x$ gets closer to $k$, with $y$ increasing or decreasing without bound. The function is undefined at $x=k$.

A graph represents a function of $x$ when every vertical line meets it at most once.

The part of the plane shown on a calculator or graphing app, set by the least and greatest values of $x$ and $y$ on the screen.

A point where a graph crosses or touches the $x$-axis, found by putting $y=0$. Its $x$-coordinate is a zero of the function.

The point where a graph crosses the $y$-axis, found by putting $x=0$. In $y=mx+c$ it is $(0,c)$; the value $c$ itself is also called the $y$-intercept.

A value of $x$ for which $f(x)=0$. The zeros of $f$ are the roots of the equation $f(x)=0$ and the $x$-coordinates of the $x$-intercepts of $y=f(x)$.