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

24 sample cards from the Polynomials, Graphs and Functions deck

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

1. 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.

2. 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.

3. 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 $a<0$).

4. 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.

5. 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 $q$.

6. 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.

7. 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.

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

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

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

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

10. 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.

11. 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 $2$.

12. 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.

13. 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.

14. 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.

15. 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.

16. Is function composition commutative? Illustrate the general rule.

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

17. 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 $f\big(f^{-1}(x)\big) = x$.

18. 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 $x$.

19. 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 $f$.

20. 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 $y = x$.

21. 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.

22. 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.

23. 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.

24. 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 $x=-3$.

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Polynomials, Graphs and Functions flashcards FAQ