Mathematics Admissions Test (MAT) Polynomials, Graphs and Functions Syllabus

How do you find the coordinates where a graph $y=f(x)$ crosses the $x$-axis and the $y$-axis?

$x$-axis intercepts (roots): set $y=0$ and solve $f(x)=0$. $y$-axis intercept: set $x=0$ and evaluate $f(0)$.

What is a turning point (stationary point) and how do you locate it?

A point where the gradient is zero, i.e. where $\frac{dy}{dx}=0$. Solving this gives the $x$-coordinates of maxima, minima or stationary points of inflection.

What is the effect of a repeated root (multiplicity) on the shape of a polynomial graph at that root?

Multiplicity 1: graph crosses the axis straight through. Even multiplicity (e.g. 2): graph touches the axis and turns back (tangent to the axis). Odd multiplicity $\geq 3$: graph crosses but flattens with a point of inflection.

How do you find the points of intersection of two curves $y = f(x)$ and $y = g(x)$?

Set $f(x) = g(x)$ and solve for $x$; substitute each solution back to find the corresponding $y$-values. The solutions are the $x$-coordinates of intersection points.

What is the algebraic condition for a line to be a tangent to a curve (using the discriminant)?

Substitute the line into the curve to form a quadratic in $x$. The line is tangent when this quadratic has a repeated root, i.e. its discriminant satisfies $b^{2} - 4ac = 0$.

If solving the simultaneous equations of a line and a curve gives a quadratic with $\Delta > 0$, $\Delta = 0$, or $\Delta < 0$, what does each mean geometrically?

$\Delta > 0$: line cuts the curve at two distinct points. $\Delta = 0$: line is a tangent (touches at one point). $\Delta < 0$: line does not meet the curve.

What is the geometric meaning of a tangent line to a curve at a point?

A straight line that touches the curve at that point and has the same gradient as the curve there; it just touches without (locally) crossing.

Describe the translation of $y=f(x)$ given by $y = f(x) + a$.

A translation by vector $\begin{pmatrix} 0 \\ a \end{pmatrix}$: the graph moves up by $a$ units (down if $a<0$).

Describe the translation of $y=f(x)$ given by $y = f(x + a)$.

A translation by vector $\begin{pmatrix} -a \\ 0 \end{pmatrix}$: the graph moves left by $a$ units (right if $a<0$). Horizontal translations act 'oppositely' to the sign inside the bracket.

What single translation vector describes $y = f(x - p) + q$?

A translation by $\begin{pmatrix} p \\ q \end{pmatrix}$: right by $p$ and up by $q$.

Describe the transformation of $y=f(x)$ to $y = a\,f(x)$ for $a>0$.

A vertical stretch by scale factor $a$ (each $y$-coordinate multiplied by $a$). Points on the $x$-axis are unchanged.

Describe the transformation of $y=f(x)$ to $y = f(ax)$ for $a>0$.

A horizontal stretch by scale factor $\dfrac{1}{a}$ (each $x$-coordinate divided by $a$). Points on the $y$-axis are unchanged.

What transformation does $y = -f(x)$ represent?

A reflection in the $x$-axis (each $y$-coordinate changes sign).

What transformation does $y = f(-x)$ represent?

A reflection in the $y$-axis (each $x$-coordinate changes sign).

When applying composite transformations, why does the order of operations matter, and how do you handle changes inside the function bracket versus outside?

Transformations outside $f$ (affecting $y$: $af(x)+b$) are applied to the $y$-values in the natural order. Transformations inside the bracket (affecting $x$) act in the reverse/opposite sense and must be undone working from the inside out. Order matters because, e.g., stretching then translating differs from translating then stretching.

Describe the sequence of transformations mapping $y = f(x)$ to $y = 2f(x - 3)$.

Translate right by $3$ (replace $x$ by $x-3$), then vertically stretch by scale factor $2$. Equivalently: a horizontal translation $\begin{pmatrix}3\\0\end{pmatrix}$ followed by a vertical stretch factor $2$.

Define the domain and the range of a function.

The domain is the set of all permitted input values ($x$). The range is the set of all output values ($f(x)$) that the function actually produces over that domain.

What is the difference between a function and a general mapping?

A function maps each element of the domain to exactly one element of the range (one-to-one or many-to-one). A mapping that sends one input to several outputs (one-to-many) is not a function.

In interval notation, what is the difference between $[a, b]$ and $(a, b)$?

$[a,b]$ is the closed interval $a \leq x \leq b$ (endpoints included). $(a,b)$ is the open interval $a < x < b$ (endpoints excluded). A square bracket includes the endpoint; a round bracket excludes it.

Given $f$ and $g$, what does the composite function $fg(x)$ (also written $f \circ g$) mean?

Apply $g$ first, then apply $f$ to the result: $fg(x) = f\big(g(x)\big)$. The inner function acts first.

Is function composition commutative? Illustrate the general rule.

No. In general $fg(x) \neq gf(x)$; the order of composition matters.

What condition must a function satisfy to have an inverse, and what does the inverse do?

It must be one-to-one (each output comes from exactly one input) on its domain. The inverse $f^{-1}$ reverses the function: $f^{-1}\big(f(x)\big) = x$ and $f\big(f^{-1}(x)\big) = x$.

What is the process for finding the inverse function $f^{-1}(x)$ algebraically?

Write $y = f(x)$, swap $x$ and $y$ (or rearrange for $x$), then solve to make $y$ the subject; the result is $f^{-1}(x)$. Often write $y=f(x)$, rearrange to get $x$ in terms of $y$, then replace $y$ with $x$.

How are the domain and range of $f$ related to those of $f^{-1}$?

They swap: the domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.

What is the geometric relationship between the graphs of $y = f(x)$ and $y = f^{-1}(x)$?

They are reflections of each other in the line $y = x$.

Define the modulus (absolute value) function $|x|$.

$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ It gives the non-negative magnitude of $x$, so $|x| \geq 0$ always.

How do you obtain the graph of $y = |f(x)|$ from the graph of $y = f(x)$?

Reflect any part of the graph that lies below the $x$-axis (where $f(x)<0$) up into the region above the $x$-axis; leave the rest unchanged. The result is never negative.

How do you obtain the graph of $y = f(|x|)$ from the graph of $y = f(x)$?

Keep the part of the graph for $x \geq 0$, then reflect it in the $y$-axis to produce the part for $x < 0$. The graph becomes symmetric about the $y$-axis.

How do you solve an equation or inequality involving a modulus, e.g. $|x - 2| = 5$ or $|x-2| < 5$?

For $|A| = b$ (with $b \geq 0$): solve $A = b$ and $A = -b$. For $|A| < b$: solve $-b < A < b$. For $|A| > b$: solve $A > b$ or $A < -b$. (Alternatively square both sides when both sides are non-negative.) E.g. $|x-2|=5 \Rightarrow x=7$ or $x=-3$.