Exam-Style Questions.

1.	GCSE Higher

The graph of the curve A with equation $y=f(x)$ is transformed to give the graph of the curve B with equation $y=5-f(x)$.

The point on A with coordinates (3, 9) is mapped to the point W on B.

Find the coordinates of W.

Worked Solution

2.	GCSE Higher

(a) By completing the square, solve $x^2+8x+13=0$ giving your answer to three significant figures.

(b) From the completed square you found in part (a) find the minimum value of the curve $y=x^2+8x+13$.

Worked Solution

3.	GCSE Higher

The graph of the following equation is drawn and then reflected in the x-axis

$$y = 2x^2 - 3x + 2$$

(a) What is the equation of the reflected curve?

The original curve is reflected in the y-axis.

(b) What is the equation of this second reflected curve?

Worked Solution

4.	GCSE Higher

(a) Find the interval for which $x^2 - 9x + 18 \le 0$

(b) The point (-4, -4) is the turning point of the graph of $y = x^2 + ax + b$, where a and b are integers. Find the values of a and b.

Worked Solution

5.	GCSE Higher

(a) Write $2x^2+8x+27$ in the form $a(x+b)^2+c$ where $a$, $b$, and $c$ are integers, by 'completing the square'

(b) Hence, or otherwise, find the line of symmetry of the graph of $y = 2x^2+8x+27$

Worked Solution

6.	IB Standard

Let $f (x)=a(x-b)^2+c$. The vertex of the graph of $f$ is at (4, -3) and the graph passes through (3, 2).

(a) Find the value of $c$.

(b) Find the value of $b$.

Worked Solution

7.	IB Standard

A function is defined as $f(x) = 2{(x - 3)^2} - 5$ .

(a) Show that $f(x) = 2{x^2} - 12x + 13$.

(b) Write down the equation of the axis of symmetry of this graph.

(d) Write down the y-intercept.

(e) Make a sketch the graph of $f(x)$.

Let $g(x) = {x^2}$. The graph of $f(x)$ may be obtained from the graph of $g(x)$ by the two transformations:

a stretch of scale factor $s$ in the y-direction;
followed by a translation of $\left( {\begin{array}{*{20}{c}} j\\ k \end{array}} \right)$ .

(f) Find the values of $j$, $k$ and $s$.

Worked Solution

8.	IB Standard

$f$ and $g$ are two functions such that $g(x)=3f(x+2)+7$.

The graph of $f$ is mapped to the graph of $g$ under the following transformations:

A vertical stretch by a factor of $a$ , followed by a translation $\begin{pmatrix}b \\c \\ \end{pmatrix}$

Find the values of

(a) $a$;

(b) $b$;

(d) Consider two other functions $h$ and $j$. Let $h(x)=-j(2x)$. The point A(8, 7) on the graph of $j$ is mapped to the point B on the graph of $h$. Find the coordinates of B.

Worked Solution

9.	IB Standard

Let $f(x)=5x^2-20x+k$. The equation $f(x)=0$ has two equal roots.

(a) Write down the value of the discriminant.

(b) Hence, show that $k=20$.

The graph of $f$ has its vertex on the x-axis.

(d) Find the coordinates of the vertex of the graph of $f$.

The function can be written in the form $f(x)=a(x-h)^2+j$.

(e) Find the value of $a$.

(f) Find the value of $h$.

(g) Find the value of $j$.

(h) The graph of a function $g$ is obtained from the graph of $f$ by a reflection in the x-axis, followed by a translation by the vector $\begin{pmatrix} 0 \\ 3 \\ \end{pmatrix} $. Find $g$, giving your answer in the form $g(x)=Ax^2+Bx+C$.

Worked Solution

10.	IB Standard

Let $f(x) = {x^2}$ and $g(x) = 3{(x+2)^2}$ .

The graph of $g$ can be obtained from the graph of $f$ using two transformations.

(a) Give a full description of each of the two transformations.

(b) The graph of $g$ is translated by the vector $ \begin{pmatrix}-4\\5\\ \end{pmatrix}$ to give the graph of $h$.

The point $( 2{\text{, }}-1)$ on the graph of $f$ is translated to the point $P$ on the graph of $h$.

Find the coordinates of $P$.

Worked Solution

11.	IB Analysis and Approaches

Part of the graph of a function, $f$ , is shown in the following diagram. The graph of $y = f(x)$ has a y-intercept at $(0, 1.5)$ , an x-intercept at $(a , 0)$ and horizontal asymptotes $y = 5$ and $y = -2$.

Consider the function $g(x) = |f(|x|)| $

(a) Sketch the graph of $y = g(x)$, labelling any axis intercepts and giving the equation of the asymptotes.

(b) Find the possible values of $k$ such that $ (g(x))^2 = k $ has exactly four solutions.

Worked Solution

12.	IB Standard

Let $f$ and $g$ be functions such that $g(x) = 3f(x - 2) + 7$ .

The graph of $f$ is mapped to the graph of $g$ under the following transformations: vertical stretch by a factor of $k$ , followed by a translation $\left( \begin{array}{l} p\\ q \end{array} \right)$ .

Write down the value of:

(a) $k$

(b) $p$

(d) Let $h(x) = - g(2x)$ . The point A($8$, $7$) on the graph of $g$ is mapped to the point ${\rm{A}}'$ on the graph of $h$ . Find ${\rm{A}}'$

Worked Solution

13.	IB Analysis and Approaches

The graph of $ y = f(|x|) $ for $ -2 \leq x \leq 2 $ is shown in the following diagram.

(a) On the following axes, sketch the graph of $ y = |f(|x|)| $ for $ -2 \leq x \leq 2 $.

It is given that $ f $ is an odd function.

(b) On the following axes, sketch the graph of $ y = f(x) $ for $ -2 \leq x \leq 2 $.

It is also given that

$$ \int_{0}^{2} f(|x|) \, dx = \dfrac{2}{3} $$

$$ \int_{-2}^{2} f(x) \, dx; $$

(d) Evaluate

$$ \int_{-2}^{2} \left( f(|x|) + f(x) \right) \, dx. $$

Worked Solution

14.	IB Standard

Two functions are defined as follows: $f(x) = 2\ln x$ and $g(x) = \ln \frac{x^2}{3}$.

(a) Express $g(x)$ in the form $f(x) - \ln a$ , where $a \in {{\mathbb{Z}}^ + }$ .

(b) The graph of $g(x)$ is a transformation of the graph of $f(x)$ . Give a full geometric description of this transformation.

Worked Solution

Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.

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