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

24 sample cards from the Coordinate Geometry and Graphs deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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 $y\geq 0$.

2. 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 $x\to-\infty$.

3. 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.

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

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

5. 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.

6. 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).

7. 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 $f(x)=g(x)$.

8. 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.

9. 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 $\begin{pmatrix}0\\a\end{pmatrix}$.

10. 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 $\begin{pmatrix}-a\\0\end{pmatrix}$.

11. 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.

12. 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.

13. 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.

14. 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.

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

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

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

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

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

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

18. 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).

19. 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 $\frac{1}{a}$.

20. 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 $(1,3)$.

21. 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 $y-y_{1}=-\frac{1}{m}(x-x_{1})$.

22. 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: $r=\frac{1}{2}\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$.

23. 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 ($a<0$).

24. 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.

What this deck covers

Coordinate Geometry and Graphs flashcards FAQ