# Exam-Style Questions.

## Problems adapted from questions set for previous Mathematics exams.

### 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.

### 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 $$y=x^2+8x+13$$.

### 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?

### 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.

### 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 $$y = 2x^2+8x+27$$

(c) Hence, or otherwise, find the turning point of the graph of $$y = 2x^2+8x+27$$

### 6.

IB Analysis and Approaches

Part of the function of $$y=f(x)$$ is shown in the following diagram.

(a) Write down the value of $$f(1)$$.

(b) Write down the value of $$ff(7)$$.

(c) Sketch the graph of $$g(x)$$ if $$g(x) = -f(x)-1$$ on the same set of axes above.

### 7.

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 $$b$$.

(c) Find the value of $$a$$.

### 8.

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.

(c) Find the coordinates of the vertex of the graph of $$f(x)$$.

(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 $$s$$.

### 9.

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) $$b$$;

(c) $$c$$.

(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.

### 10.

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.

(c) Write down the solution of $$f(x)=0$$.

(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 $$g(x)=Ax^2+Bx+C$$.

### 11.

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 $$P$$.

### 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) $$p$$

(c) $$q$$

(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 $${\rm{A}}'$$

### 13.

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.

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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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