Test of Mathematics for University Admission (TMUA) Coordinate Geometry and Graphs Syllabus

To find where a line meets a circle, what method do you use and what does the discriminant tell you?

Substitute the line equation into the circle equation to get a quadratic in one variable. The discriminant $b^{2}-4ac$: $>0$ two intersection points (secant), $=0$ one point (tangent), $<0$ none (line misses circle).

How do you find the intersection points of two curves/lines algebraically?

Solve their equations simultaneously (by substitution or elimination); each solution pair $(x,y)$ is an intersection point. For a line and curve, substitute to form one equation in a single variable.

What does the discriminant of the resulting quadratic indicate when intersecting a line with a curve?

$b^{2}-4ac>0$: two intersections; $=0$: a single (tangential) intersection; $<0$: no real intersections.

Describe the shape and key features of the graph $y=x^{2}$.

An upward-opening parabola, symmetric about the $y$-axis, with a minimum (vertex) at the origin $(0,0)$.

Describe the shape of the cubic graph $y=x^{3}$.

An increasing curve passing through the origin with a point of inflection at $(0,0)$; it goes from bottom-left to top-right ($y\to-\infty$ as $x\to-\infty$, $y\to+\infty$ as $x\to+\infty$).

Describe the graph of the reciprocal function $y=\frac{1}{x}$, including its asymptotes.

A rectangular hyperbola in two branches (quadrants 1 and 3). Asymptotes are the $x$-axis ($y=0$) and the $y$-axis ($x=0$); it is undefined at $x=0$.

What is the shape of $y=\sqrt{x}$ and its domain?

The upper half of a sideways parabola, starting at the origin and increasing; domain $x\geq 0$ and range $y\geq 0$.

Sketch features of an exponential graph $y=a^{x}$ with $a>1$.

Always positive, increasing, passes through $(0,1)$, with the $x$-axis ($y=0$) as a horizontal asymptote as $x\to-\infty$.

How do you find the $x$-intercepts (roots) of a curve $y=f(x)$?

Set $y=0$ and solve $f(x)=0$. The solutions are the $x$-coordinates where the curve crosses the $x$-axis.

How do you find the $y$-intercept of a curve $y=f(x)$?

Set $x=0$ and evaluate $f(0)$; the point is $(0,f(0))$.

What is a vertical asymptote and how do you locate it for a rational function?

A vertical line $x=k$ that the curve approaches but never meets. For a rational function it occurs where the denominator equals zero (and the numerator does not), i.e. the function is undefined.

How do you find the horizontal asymptote of a rational function as $x\to\pm\infty$?

Compare degrees of numerator and denominator: if numerator degree < denominator, asymptote is $y=0$; if equal, $y=\frac{\text{leading coeff numerator}}{\text{leading coeff denominator}}$; if numerator is higher, there is no horizontal asymptote (possibly an oblique one).

How can the equation $f(x)=g(x)$ be solved graphically?

Sketch $y=f(x)$ and $y=g(x)$ on the same axes; the $x$-coordinates of their intersection points are the solutions of $f(x)=g(x)$.

How is the number of real solutions of $f(x)=0$ read from the graph of $y=f(x)$?

It equals the number of times the curve crosses (or touches) the $x$-axis; a touch point corresponds to a repeated root.

Describe the transformation $y=f(x)+a$ for $a>0$.

A vertical translation: the graph of $y=f(x)$ moves up by $a$ units. Translation vector $\begin{pmatrix}0\\a\end{pmatrix}$.

Describe the transformation $y=f(x+a)$ for $a>0$.

A horizontal translation: the graph moves left by $a$ units (the inside change acts in the opposite/counter-intuitive direction). Translation vector $\begin{pmatrix}-a\\0\end{pmatrix}$.

What single transformation does $y=f(x-3)+2$ apply to $y=f(x)$?

A translation by the vector $\begin{pmatrix}3\\2\end{pmatrix}$: $3$ units right and $2$ units up.

Describe the transformation $y=a\,f(x)$ for $a>1$.

A vertical stretch by scale factor $a$: each $y$-coordinate is multiplied by $a$. Points on the $x$-axis stay fixed.

Describe the transformation $y=f(ax)$ for $a>1$.

A horizontal stretch by scale factor $\frac{1}{a}$ (a compression). Each $x$-coordinate is divided by $a$; points on the $y$-axis stay fixed.

What does $y=f\!\left(\tfrac{1}{2}x\right)$ do to the graph of $y=f(x)$?

A horizontal stretch with scale factor $2$ (since the factor is $\frac{1}{a}$ with $a=\frac{1}{2}$): the graph is stretched away from the $y$-axis, doubling $x$-coordinates.

Describe the reflection $y=-f(x)$.

A reflection in the $x$-axis: each $y$-coordinate changes sign; the graph flips top-to-bottom.

Describe the reflection $y=f(-x)$.

A reflection in the $y$-axis: each $x$-coordinate changes sign; the graph flips left-to-right.

What transformation is $y=-f(-x)$?

A reflection in both axes, equivalent to a rotation of $180^{\circ}$ about the origin.

For combined transformations affecting $x$ (inside the function), in what order are they applied?

Inside (horizontal) changes act in the reverse/opposite sense and are applied 'inside-out'. For example $y=f(2x-4)=f(2(x-2))$ is a horizontal stretch factor $\frac{1}{2}$ then a translation $2$ right (translate after factorising).

Compare how vertical and horizontal transformations behave with respect to intuition.

Vertical transformations (outside $f$) act intuitively/normally: $+a$ up, $\times a$ stretches by $a$. Horizontal transformations (inside $f$) act in the opposite/counter-intuitive way: $+a$ moves left, $\times a$ compresses by factor $\frac{1}{a}$.

Describe the sequence of transformations taking $y=x^{2}$ to $y=(x-1)^{2}+3$.

Translate $1$ unit right then $3$ units up — i.e. translation by $\begin{pmatrix}1\\3\end{pmatrix}$; the vertex moves from $(0,0)$ to $(1,3)$.

How do you find the line through a given point that is perpendicular to a given line $y=mx+c$?

Use gradient $-\frac{1}{m}$ and the given point in $y-y_{1}=-\frac{1}{m}(x-x_{1})$.

What is the length of the diameter of a circle in terms of its radius, and how is the radius found from two endpoints of a diameter?

Diameter $=2r$. Given diameter endpoints, the centre is their midpoint and the radius is half the distance between them: $r=\frac{1}{2}\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$.

How can completing the square on a quadratic $y=ax^{2}+bx+c$ reveal its vertex for sketching?

Write $y=a(x-h)^{2}+k$; the vertex (turning point) is at $(h,k)$ and the axis of symmetry is $x=h$. The sign of $a$ shows whether it is a minimum ($a>0$) or maximum ($a<0$).

What is the effect on roots/intercepts when a graph $y=f(x)$ is reflected in the $x$-axis to give $y=-f(x)$?

The $x$-intercepts (roots) are unchanged, because where $f(x)=0$ we also have $-f(x)=0$; only the $y$-values between roots are reflected.