# Exam-Style Questions on Graphs

## Problems on Graphs adapted from questions set in previous exams.

### 1.

GCSE Higher

The equation of the line L1 is $$y = 2 - 5x$$.

The equation of the line L2 is $$3y + 15x + 17 = 0$$.

Show that these two lines are parallel.

### 2.

GCSE Higher

Show that line $$5y = 7x - 7$$ is perpendicular to line $$7y = -5x + 55$$.

### 3.

GCSE Higher

A straight line goes through the points $$(a, b)$$ and $$(c, d)$$, where

$$a + 3 = c$$ $$b + 6 = d$$

Find the gradient of the line.

### 4.

GCSE Higher

The graph shows the height of water in a container over a period time during which the water enters the container at a constant rate.

Which of the following might be a diagram of the container?

a. b. c. d. e.

### 5.

GCSE Higher

The diagram is of a container which is filled with water entering at a constant rate.

Which of the following might be the graph of height of the water in the container plotted against time?

a. b. c. d. e.

### 6.

GCSE Higher

Match the equation with the letter of its graph

EquationGraph
$$y=3-\frac{10}{x}$$
$$y=2^x$$
$$y=\sin x$$
$$y=x^2+7x$$
$$y=x^2-8$$
$$y=2-x$$

### 7.

GCSE Higher

The graph of y = f(x) is drawn accurately on the grid.

(a) Write down the coordinates of the turning point of the graph.

(b) Write down estimates for the roots of f(x) = 0

(c) Use the graph to find an estimate for f(-5.5).

### 8.

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

### 9.

GCSE Higher

Suppose a rhombus ABCD is drawn on a coordinate plane with the point A situated at (4,7). The diagonal BD lies on the line $$y = 2x - 5$$

Find the equation the line that passes through A and C.

### 10.

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?

### 11.

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.

### 12.

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

### 13.

IB Studies

Consider a straight line graph L1, which intersects the x-axis at A(8, 0) and the y-axis at B (0, 4).

(a) Write down the coordinates of C, the midpoint of line segment AB.

(b) Calculate the gradient of the line L1.

The line L2 is parallel to L1 and passes through the point (5 , 9).

(c) Find the equation of L2. Give your answer in the form $$ay + bx + c = 0$$ where $$a, b \text{ and } c \in \mathbb{Z}$$.

### 14.

IB Standard

Let $$f(x) = \frac{9x-3}{bx+9}$$ for $$x \neq -\frac5b, b \neq 0$$.

(a) The line $$x = 3$$ is a vertical asymptote to the graph of $$f$$. Find the value of b.

(b) Write down the equation of the horizontal asymptote to the graph of $$f$$.

(c) The line $$y = c$$ , where $$c\in \mathbb R$$ intersects the graph of $$\begin{vmatrix}f(x) \end{vmatrix}$$ at exactly one point. Find the possible values of $$c$$.

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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