2 Functions

A straight-line model can be written y=mx+c, where m is the gradient and c is the y-intercept. The gradient measures change in y per unit change in x.

Use the form that matches the data: two points give m=(y₂−y₁)/(x₂−x₁); a point and a gradient give y−y₁=m(x−x₁). Check the scale and units before interpreting m.

Through (2,5) and (6,13), m=8/4=2, so y−5=2(x−2), or y=2x+1. At x=0 the model predicts 1, not 0.

A negative gradient means y falls as x rises; it does not mean the intercept is negative. Do not swap coordinates or treat correlation as proof that the line is causal.

Line-form map: $y=mx+c$ shows gradient and $y$-intercept, $y-y_1=m(x-x_1)$ uses a known point, and $ax+by+d=0$ is general form. Set $y=0$ for the $x$-intercept and $x=0$ for the $y$-intercept. Parallel non-vertical lines have $m_1=m_2$; perpendicular non-vertical lines satisfy $m_1m_2=-1$. Vertical lines $x=k$ require a separate case because their gradient is undefined.

The inverse function f⁻¹ reverses the mapping: f⁻¹(f(x))=x and f(f⁻¹(x))=x on their appropriate domains. A function needs to be one-to-one for an inverse function to exist without restricting the domain.

To find an inverse, write y=f(x), interchange x and y, then solve for y. Graphically, f and f⁻¹ reflect in y=x, so the domain and range swap.

For f(x)=2x+3, y=2x+3 gives x=(y−3)/2, so f⁻¹(x)=(x−3)/2. A quadratic needs a domain restriction before its inverse is a function.

f⁻¹(x) is not 1/f(x). Always check composition and state the domain restriction when the original graph fails the horizontal-line test.

A graph shows how outputs depend on inputs. A useful sketch preserves key behaviour—intercepts, turning points, asymptotes, end behaviour and domain—not every plotted pixel.

Start with the domain and scale, then find features that determine the shape. Use a calculator or points to check the sketch, but interpret the graph rather than copying an unlabelled screen.

For y=(x−2)²−1, the vertex is (2,−1), the x-intercepts are 1 and 3, and the parabola opens upward. Those three facts are more informative than a dense table of values.

A sketch need not be to scale, but it must not invent features. Distinguish an intercept from a turning point and do not ignore a restricted domain.

A 'draw' should be accurately plotted, while a 'sketch' preserves essential features. Label both axes, units and key coordinates. Technology can graph $f+g$ or $f-g$ pointwise: if $f(x)=x^2$ and $g(x)=2x$, then $(f-g)(x)=x^2-2x$; its zeros at 0 and 2 are where the original graphs intersect because $f(x)=g(x)$.

Key graph features translate a picture into statements: intercepts show zeros or initial values, a gradient shows rate of change, a turning point marks a local extreme, and an asymptote describes a value the graph approaches.

Read the axes and units first. Then connect the feature to the context: a maximum may be a capacity or peak, while a negative gradient may represent decline. A graph can show association without proving why it occurs.

If a concentration curve levels off at 8 mg L⁻¹, the horizontal tendency suggests a limiting value; it does not prove the process has stopped or that 8 is a universal constant.

Do not confuse a horizontal tangent with a horizontal asymptote. Check whether the statement concerns one point, a local interval or long-run behaviour.

Feature checklist: $x$-intercepts are roots or zeros, the $y$-intercept is $f(0)$, and a quadratic vertex lies on its axis of symmetry. Record local maxima/minima, vertical and horizontal asymptotes, and any graph symmetry. To find the intersection of two curves with technology, graph both and solve $f(x)=g(x)$, then report the common point and verify it satisfies both equations.

A mathematical model uses a function to represent a real relationship. The model is useful only over a stated context and with assumptions about variables, units and measurement.

Choose a form from behaviour: constant change suggests linear, repeated proportional change suggests exponential, and a single peak may suggest a quadratic. Fit parameters from data, then test residuals or predictions and state the domain where the model is credible.

A population rising by roughly 4% per year may be modelled by P(t)=P₀(1.04)^t. The model predicts a value, not a biological law; resource limits can make the extrapolation fail.

A close fit does not validate an unlimited extrapolation. Explain why the chosen form matches the observed mechanism and where the assumption breaks.

Syllabus model map: linear and piecewise-linear models represent constant rates or changing tariffs; quadratics use roots, vertex and symmetry; exponentials use a repeated factor and horizontal asymptote; $f(x)=ax^n$ models direct or inverse variation and may have a vertical asymptote when $n<0$; cubics allow two turns; sinusoids $a\sin(bx)+d$ or $a\cos(bx)+d$ have amplitude $|a|$, period $360^\circ/|b|$ and principal axis $y=d$. Use technology for roots and parameters, then interpret them in context.

To model a situation, define variables and units, choose a functional form, estimate parameters, compare predictions with evidence and revise or qualify the model.

The context determines what counts as a reasonable assumption. Keep a train/test or observed/predicted distinction when possible, and inspect whether errors are random or systematic. A model that fits one interval may fail outside it.

A delivery-time model based on distance may work for local routes but overpredict long routes if motorway travel changes the relationship. Refit or split the model rather than hiding the systematic error.

The calculator selecting a curve is not the modelling process. The final claim must include assumptions, units, validation evidence and a domain of use.

The composite (f∘g)(x)=f(g(x)) applies g first and then f. Its domain is restricted wherever the inner output is outside the outer function's domain.

Keep the order visible. If f(x)=√x and g(x)=x−1, then (f∘g)(x)=√(x−1), so x≥1; reversing the order gives √x−1 with a different domain and meaning.

For f(x)=2x+1 and g(x)=x², (f∘g)(3)=2·9+1=19, while (g∘f)(3)=7²=49. The functions need not commute.

Composition is not multiplication. Check which function acts first and state the domain rather than assuming both orders are valid.

Inverse workflow: restrict the original domain until the function is one-to-one, write $y=f(x)$, swap $x$ and $y$, and solve for $y$. Then $(f\circ f^{-1})(x)=x$ on the range of $f$ and $(f^{-1}\circ f)(x)=x$ on the restricted domain. For $f(x)=(x-3)^2$ restricted to $x\ge3$, $f^{-1}(x)=3+\sqrt{x}$; choosing $x\le3$ would give the other branch $3-\sqrt{x}$.

For y=f(x), adding outside the function changes outputs: y=f(x)+b shifts vertically. Changing the input changes where an existing feature occurs: y=f(x−a) shifts right by a.

A factor outside, y=af(x), stretches or reflects vertically; a factor inside, y=f(bx), changes horizontal scale by 1/|b| and may reflect when b<0. Composite transformations must be applied in the stated order.

From y=x², the graph y=2(x−3)²+1 has vertex (3,1), opens upward and is twice as steep vertically. Read the vertex after the horizontal shift, then the vertical changes.

Inside and outside signs behave differently: f(x−3) moves right, while f(x)+3 moves up. Do not reverse a horizontal scale factor.

AHL models extend the SL families: half-life uses exponential decay; $a+b\ln x$ models logarithmic change; $a\sin(b(x-c))+d$ models cycles; $L/[1+Ce^{-kx}]$ models growth limited by carrying capacity $L$; piecewise functions use different rules on stated intervals.

For a half-life $h$, $Q(t)=Q_0(1/2)^{t/h}$. In a sinusoidal model, radians are assumed unless degrees are marked, the period is $2\pi/|b|$, and $c$ is a horizontal translation or phase shift. In a logistic model, $L$ is the horizontal asymptote and carrying capacity.

Example

If a quantity starts at 800 and has half-life 3 years, $Q(t)=800(1/2)^{t/3}$, so $Q(6)=200$. For a piecewise model, solve shared-boundary parameter conditions when continuity is required; the formal definition of continuity is not required.

A model family is chosen from mechanism and shape, not fit alone. Do not use degrees in a radian model, call $L$ the initial value of a logistic curve, or assume two piecewise rules join continuously without checking their boundary values.

A logarithmic axis spaces equal ratios equally. Taking logs can turn a multiplicative model into a linear one: y=Ax^k gives log y=log A+k log x, while y=Ab^x gives log y=log A+x log b.

The slope on a log–log plot estimates a power k; the slope on a semi-log plot estimates log b. Choose the transformation that matches the proposed mechanism and keep the same log base when interpreting intercepts.

If doubling x roughly quadruples y, a log–log slope near 2 is more informative than a straight-line fit on ordinary axes. The intercept gives the scale factor only after undoing the logarithm.

A straight line after transformation does not prove the original relationship. Check units, zero/negative values and back-transform predictions before making the contextual claim.

Logarithms compress very large or small positive values into a manageable scale. On a semi-log plot, linearity supports $y=Ab^x$ and slope is $\log b$; on a log–log plot, linearity supports $y=Ax^k$ and slope is $k$, with intercept $\log A$. Students interpret these graphs in examinations but are not required to draw or sketch them. Zero and negative values cannot be logged without redefining the model.