Maths AI terms in Functions

Half the vertical distance between the maximum and minimum values of a periodic function; for $a\sin (bx)+d$ and $a\cos (bx)+d$ it equals $\mathrm{\mid }a\mathrm{\mid }$.

The angle $\theta$ between a line or incline and the horizontal, measured upwards; it is related to the gradient $m$ by $\tan \theta =m$.

A statement accepted as true, without proof, in order to simplify a real situation enough for it to be modelled mathematically.

A straight line that the graph of a function approaches more and more closely, getting as near to it as desired, as $x$ or $y$ becomes very large in size.

The vertical line about which a graph is symmetrical; for the parabola $y=a{x}^2+bx+c$ it is the line $x=-\frac{b}{2a}$.

A level of output at which total revenue equals total cost, so the profit is zero; graphically, a point of intersection of the revenue and cost graphs.

The limiting value $L$ of a logistic model $f(x)=\frac{L}{1+C{\mathrm{e}}^{-kx}}$, which is the horizontal asymptote $y=L$ that the output approaches but does not reach; in a population model it is the largest population the environment can support.

The function obtained by applying one function to the output of another; the composite $f\circ g$ is defined by $(f\circ g)(x)=f(g(x))$, so $g$ is applied first and $f$ second.

A transformation made by applying two or more transformations one after another, in a stated order.

The constant $a$ in a variation relationship $y=a{x}^n$, equal to $\frac{y}{x^n}$ for every pair of corresponding values.

A function whose graph can be drawn without lifting the pen from the paper; a piecewise function is continuous at a boundary when the formulae on either side give the same value there.

A function that gives the total cost of producing or buying a quantity of goods, usually a fixed cost plus a cost for each unit.

A model of the form $f(x)=a{x}^3+b{x}^2+cx+d$, where $a\mathrm{\ne }0$.

The positive constant $k$ in an exponential decay model $N(t)=N_0{\mathrm{e}}^{-kt}$; it measures how fast the quantity decreases, as a proportion of its current value per unit of time.

The output of a function, whose value depends on the value of the independent variable, usually plotted on the vertical axis.

The function $f-g$ whose output at each input is the output of $f$ minus the output of $g$, so $(f-g)(x)=f(x)-g(x)$ for every $x$ in both domains.

A relationship $y=a{x}^n$ with $n$ a positive integer, in which $y$ is proportional to the $n$th power of $x$, written $y\propto x^n$.

The set of all input values for which a function is defined.

The replacement of a function's domain by a smaller set of inputs, usually chosen so that the function becomes one-to-one and has an inverse.

To represent a function by an accurate, labelled graph made in pencil, on axes with a stated scale, with points correctly plotted and joined by a ruled straight line or a smooth curve.

Change in which a quantity is multiplied by the same factor between 0 and 1 over every interval of equal length, so that it decreases by a constant percentage of its current size.

Change in which a quantity is multiplied by the same factor greater than 1 over every interval of equal length, so that it increases by a constant percentage of its current size.

Using a model to predict a value for an input that lies outside the range of the input values used to build or test the model; such predictions rely on the pattern continuing and are often unreliable.

Finding values of a model's parameters so that the model agrees with given conditions or data.

A rule that assigns to each element of a set of inputs exactly one output.

A diagram that represents a function as a box with an input arrow and an output arrow, showing the rule that turns each input into its output.

The notation $f(x)$ for the output of the function $f$ when its input is $x$.

The equation of a straight line written as $ax+by+d=0$, where $a$, $b$ and $d$ are constants, usually integers, and $a$ and $b$ are not both zero.

The steepness of a straight line, equal to the vertical change divided by the horizontal change between any two points on the line.

The formula $m=\frac{y_2-y_1}{x_2-x_1}$, which gives the gradient $m$ of the straight line through the points $(x_1,y_1)$ and $(x_2,y_2)$, where $x_1\mathrm{\ne }{x}_2$.

The equation of a straight line written as $y=mx+c$, where $m$ is the gradient and $c$ is the y-coordinate of the y-intercept.

The set of all points $(x,y)$ in the plane with $x$ in the domain of the function and $y=f(x)$.

A solution of an equation $f(x)=g(x)$ found as the $x$-coordinate of a point of intersection of the graphs of $y=f(x)$ and $y=g(x)$.

A horizontal line $y=b$ that a graph approaches as $x$ becomes very large and positive, or very large and negative, so that the outputs get closer and closer to $b$.

A function is one-to-one, and so has an inverse function, if and only if no horizontal line meets its graph more than once.

A transformation that multiplies the $x$-coordinate of every point by a scale factor, leaving the $y$-coordinate unchanged; the graph of $y=f(qx)$ is a horizontal stretch of $y=f(x)$ with scale factor $\frac{1}{q}$.

The function that maps every input to itself, $x\mapsto x$; its graph is the line $y=x$.

The point, line or graph that results when a transformation is applied to an original point, line or graph.

The input of a function, whose values are chosen or given, usually plotted on the horizontal axis.

The value of a model's output when the input is 0; for $f(x)=k{a}^x+c$ it equals $k+c$.

Using a model to predict a value for an input that lies within the range of the input values used to build or test the model.

A way of writing a set of real numbers between two endpoints, with $[a,b]$ including both endpoints, $]a,b[$ excluding both, and $[a,b[$ or $]a,b]$ including just one.

A line that is mapped onto itself by a transformation; if every point on the line stays fixed, it is a line of invariant points.

A point that is mapped to itself by a transformation, so that its coordinates are unchanged.

The function $f^{-1}$ that reverses the effect of a one-to-one function $f$, so that if $f(a)=b$ then $f^{-1}(b)=a$.

A relationship in which a quantity is inversely proportional to the square of the distance from its source, $I=\frac{k}{d^2}$.

A relationship $y=\frac{a}{x^n}$ with $n$ a positive integer (equivalently $y=a{x}^{-n}$), in which $y$ is proportional to the reciprocal of the $n$th power of $x$, written $y\propto \frac{1}{x^n}$.

A point, line or behaviour of a graph that describes the function and must be labelled on a sketch, such as an intercept, a maximum or minimum point, an asymptote, a point of intersection or an end point of the domain.

A feature of the real situation that a model ignores or represents poorly, or a range of inputs where the model gives unreliable results.

A function whose graph is a straight line, with an equation of the form $f(x)=mx+c$, where $m$ and $c$ are constants.

A model of the form $f(x)=mx+c$, in which the output changes by the same amount $m$ for every increase of 1 in the input.

Transforming one or both variables of a data set, usually by taking logarithms, so that the relationship between the transformed variables is a straight line whose gradient and intercept give the parameters of the original model.

A point on a graph whose $y$-coordinate is greater than or equal to the $y$-coordinates of all nearby points on both sides of it.

A point on a graph whose $y$-coordinate is less than or equal to the $y$-coordinates of all nearby points on both sides of it.

A graph in which both axes are logarithmic, or equivalently a graph of $\ln y$ against $\ln x$ (or $\log y$ against $\log x$); power relationships appear on it as straight lines whose gradient is the power.

A model of the form $f(x)=\frac{L}{1+C{\mathrm{e}}^{-kx}}$, where $L,C,k>0$, describing growth that is close to exponential at first and then slows, so that the output approaches the limiting value $L$.

A function in which at least one output comes from two or more different inputs.

A pairing of the elements of one set (the inputs) with elements of another set (the outputs), often shown as a diagram of arrows from each input to its output.

The price and quantity at which the quantity buyers want to buy equals the quantity sellers want to sell; graphically, the point of intersection of the demand and supply curves.

A function, equation or graph that represents a real situation in mathematical form, so that it can be analysed and used to make predictions.

The repeated process of stating assumptions about a real situation, choosing and fitting a model, testing it against reality, reflecting on and adapting it, and finally using it to explain, predict or inform decisions.

A model of the form $f(x)=a+b\ln x$, where $a$ and $b$ are constants and $x>0$; its graph has the vertical asymptote $x=0$ and increases (for $b>0$) or decreases (for $b<0$) ever more slowly.

For a non-zero number $m$, the number $-\frac{1}{m}$, found by turning $m$ upside down and changing its sign; it is the gradient of any line perpendicular to a line of gradient $m$.

A function in which every output comes from exactly one input, so that different inputs always give different outputs.

The symmetrical U-shaped curve that is the graph of a quadratic function; it opens upwards when $a>0$ and downwards when $a<0$.

Lines in the same plane that never meet; two non-vertical lines are parallel exactly when their gradients are equal, $m_1=m_2$.

A constant in the formula of a model that is fixed for a particular situation but can take different values in different situations.

The gradient of an incline written as a percentage, equal to the vertical rise divided by the horizontal run, multiplied by 100%.

The horizontal length of one complete cycle of a periodic function; for $a\sin (bx)+d$ and $a\cos (bx)+d$ with $x$ in degrees it equals $\frac{{360}^{\circ }}{b}$.

Lines that meet at a right angle; two lines with gradients $m_1$ and $m_2$, neither vertical, are perpendicular exactly when $m_1\times m_2=-1$.

The horizontal translation $c$ of a sinusoidal model $f(x)=a\sin (b(x-c))+d$; the graph of $a\sin (bx)+d$ is moved $c$ units to the right when $c>0$ and $\mathrm{\mid }c\mathrm{\mid }$ units to the left when $c<0$.

A function defined by two or more formulae, each applying on its own interval of the domain, where the intervals do not overlap.

A model made of two or more linear functions, each applying on its own interval of the domain.

To mark the positions of given or calculated points accurately on a pair of axes.

A point that lies on two or more graphs at once; its coordinates satisfy the equations of all of them.

The equation of a straight line written as $y-y_1=m(x-x_1)$, where $m$ is the gradient and $(x_1,y_1)$ is a known point on the line.

A relationship of the form $y=a{x}^n$, where $a$ and $n$ are constants, so that multiplying $x$ by any factor $m$ multiplies $y$ by $m^n$.

The horizontal line $y=d$ about which a sinusoidal graph oscillates, midway between its maximum and minimum values.

A function $P(x)$ giving the income left after costs when $x$ items are made and sold, equal to revenue minus cost, $P(x)=R(x)-C(x)$; a negative value is a loss.

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

The set of all output values that a function takes when the input runs through the whole of its domain.

The set of input values for which a model makes sense in its real context, given by physical limits, by the meaning of the variables and by the range of the data used to build it.

A transformation that maps every point to its mirror image in a fixed line, so that the line is the perpendicular bisector of the segment joining each point to its image.

A function $R(x)$ that gives the total income from selling $x$ items; at a fixed price $p$ per item, $R(x)=px$.

The size of the step represented by each marked division along an axis, which fixes how far apart given values appear.

The number by which a stretch or enlargement multiplies distances; for a stretch these are distances from its invariant line, which a scale factor greater than 1 increases and a scale factor between 0 and 1 reduces.

A function that is equal to its own inverse, so that $f^{-1}(x)=f(x)$ and $(f\circ f)(x)=x$ for every $x$ in its domain.

A graph in which one axis (usually the vertical) is logarithmic and the other is linear, or equivalently a graph of $\ln y$ (or $\log y$) against $x$; exponential relationships appear on it as straight lines.

A model of the form $f(x)=a\sin (bx)+d$ or $f(x)=a\cos (bx)+d$, used for quantities that repeat in a regular cycle.

To represent a function by a graph, labelled as appropriate, that gives a general idea of its shape and shows its relevant features, without needing to be drawn to scale.

A control in graphing software that changes the value of a parameter continuously, redrawing the graph at once so that the effect of the parameter on its key features can be seen.

A continuous curve drawn freehand through plotted points without corners, gaps or straight-line segments, following the gradual bending of the function between the points.

The function $f+g$ whose output at each input is the sum of the outputs of $f$ and $g$, so $(f+g)(x)=f(x)+g(x)$ for every $x$ in both domains.

A table that lists chosen inputs $x$ of a function alongside the corresponding outputs $f(x)$, giving points that lie on its graph.

A rule that moves every point of a graph or shape to a new position, such as a translation, a reflection or a stretch.

A transformation that moves every point of a graph the same distance in the same direction, without changing its shape, size or orientation.

A column vector $\left(\begin{array}{c}a\\ b\end{array}\right)$ that describes a translation of $a$ units parallel to the $x$-axis and $b$ units parallel to the $y$-axis.

A point where a graph changes from rising to falling (a local maximum) or from falling to rising (a local minimum).

The extent to which a model's assumptions, type of function and outputs match the real situation over the inputs where it is used, so that conclusions drawn from it can be trusted.

The turning point of a parabola, where it meets its axis of symmetry; it is the minimum point when $a>0$ and the maximum point when $a<0$.

A vertical line $x=a$ that a graph approaches while its outputs grow without limit (positive or negative) as $x$ gets closer to $a$; the function is undefined at $x=a$.

A graph represents a function of $x$ if and only if no vertical line meets the graph more than once.

A transformation that multiplies the $y$-coordinate of every point by a scale factor $p$, leaving the $x$-coordinate unchanged; the graph of $y=pf(x)$ is a vertical stretch of $y=f(x)$ with scale factor $p$.

The rectangular part of the coordinate plane displayed by a GDC or graphing software, set by the minimum and maximum $x$-values and the minimum and maximum $y$-values.

The point where a graph crosses the x-axis, found by substituting $y=0$ and solving for $x$.

The point where a graph crosses the y-axis, found by substituting $x=0$; its y-coordinate is often called the y-intercept as well.