# Exam-Style Questions on Graph

## Problems on Graph adapted from questions set in previous Mathematics exams.

### 1.

GCSE Higher

The diagram shows part of the graph $$y=x^2-3x+6$$.

(a) By drawing a suitable straight line, use your graph to find estimates for the solutions of $$x^2 - 4x + 2 = 0$$ to one decimal place.

(b) A is the point (2,4). Calculate an estimate for the gradient of the graph at the point A.

### 2.

IB Standard

Let $$f(x)=\frac{3x}{x-q}$$, where $$x \neq q$$.

(a) Write down the equations of the vertical and horizontal asymptotes of the graph of $$f$$.

The vertical and horizontal asymptotes to the graph of $$f$$ intersect at the point Q(1, 3).

(b) Find the value of q.

(c) The point P(x, y) lies on the graph of $$f$$. Show that PQ = $$\sqrt{(x-1)^2+(\frac{3}{x-1})^2}$$

(d) Hence find the coordinates of the points on the graph of $$f$$ that are closest to (1, 3).

### 3.

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 $$s$$.

### 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.

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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