# Exam-Style Questions on Transformation of Graphs

## Problems on Transformation of Graphs adapted from questions set in 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

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?

### 3.

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.

### 4.

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 $$P$$.

### 5.

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

### 6.

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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