Question Bank | Unit 3E - Algebra and Graphs

1. The graph of $y = x^{2} - 5 x + 5$ is drawn for values of $x$ between 0 and 5. On each axis 1 cm represents 1 unit.

(a) Use the graph to estimate, correct to one decimal place, the solutions of the equation $x^{2} - 5 x + 5 = 0$ .

(b) By drawing the tangent at the point (1, 1), estimate the gradient of the curve at this point.

2. The curve $y = (x + 3) (x - 2)$ cuts the $x$ -axis at the points $A$ and $C$ , and the $y$ -axis at $B$ .

(a) Write down the coordinates of the points $A$ , $B$ and $C$ .

(b) Find the equation of the line of symmetry of the curve.

3. On the axes in the answer space, sketch and clearly label the graphs of

(a) $x + y = 0$ ,

(b) $y = x^{3}$ .

4. The curve $y = p x^{3} + 8$ intersects the $y$ -axis at $A$ and passes through the point (−2, −32).

(a) Write down the coordinates of $A$ .

(b) Calculate the value of $p$ .

5. Answer the whole of this question on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = 2 x^{2} - 5 x - 3$ and some corresponding values are given in the following table.

x	−2	−1	0	$\frac{1}{2}$	1	2	3	4	5
y	a	4	−3	b	−6	−5	0	9	22

Calculate the values of $a$ and $b$ .

Taking 2 cm to represent 1 unit on the $x$ -axis and 2 cm to represent 5 units on the $y$ -axis, draw the graph of $y = 2 x^{2} - 5 x - 3$ for the range $- 2 \leq x \leq 5$ .

(a) From your graph find

(i) the least value of $y$ ,

(ii) the values of $x$ when $y = 5$ .

(b) Find, by drawing a tangent, the gradient of the graph at the point where $x = 2$ .

(c) By drawing a suitable straight line on the same axes, use your graph to find the solutions of the equation $2 x^{2} - 5 x - 3 = 3 x + 4$ .

6. Answer the whole of this question on a single sheet of graph paper, putting the answers to parts (a) to (e) on the plain side of the paper.

In the diagram, $O$ is the origin and $B P A$ is a straight line of gradient $m$ .

(a) Given that $A$ is the point ( $a$ , 0) and $P$ is the point (2, 1), express $m$ in terms of $a$ .

(b) Show that $a = 2 - \frac{1}{m}$ .

(c) Given that $B$ is the point (0, $b$ ), express $b$ in terms of $m$ .

(d) Hence show that if $O A + O B = S$ , then $S = 3 - 2 m - \frac{1}{m}$ .

(e) Corresponding values of $m$ and $S$ are given in the following table.

m	−4	−3	−2	−1	−0·5	−0·25
S	11·3	9·3	7·5	6	6	7·5

Using a horizontal scale of 4 cm to 1 unit for $m$ and a vertical scale of 2 cm to 1 unit for $S$ .

Plot these values on graph paper and draw a smooth curve through your points.

(f) Use your graph to estimate the smallest value of $S$ .

7. Part of the graph of $y = \frac{16}{x^{2}}$ is drawn in the answer space.

(a) Add to this drawing the two lines of symmetry.

(b) Given that the point ( $h, - \frac{4}{3}$ ) lies on the curve, calculate the value of $h$ .

(c) A reflection maps the point (2, 8) onto the point (−8, −2).

Write down the matrix which represents this reflection.

8. The diagram shows the graphs of $y = 3 + 13 x - 4 x^{2}$ and $y = 4 x - 9$ for $- 2 \leq x \leq 5$ . They intersect at $P$ and $Q$ . Use the diagram to answer the following questions. Do not redraw the graphs.

(a) Estimate, correct to one decimal place, the solutions of the equations

(i) $3 + 13 x - 4 x^{2} = 0$ ,

(ii) $3 + 13 x - 4 x^{2} = - 10$ .

(b) Write down the equation in $x$ which has the $x$ -coordinates of $P$ and $Q$ for its solutions, and express it as simply as possible.

(c) By drawing a tangent, find the gradient of the graph $y = 3 + 13 x - 4 x^{2}$ at the point $P$ .

(d) Estimate the area between the graph $y = 3 + 13 x - 4 x^{2}$ , the $x$ -axis and the lines $x = 0$ and $x = 2$ .

(e) State the range of values of $x$ for which $x > 0$ and $4 x - 9 > 3 + 13 x - 4 x^{<2}$ .

9. On the axes in the answer space, sketch, and label clearly, the graphs of

(a) $y = x + 1$ ,

(b) $y = 1 - x^{2}$ .

10. Answer the whole of this question on a sheet of graph paper. The following is an incomplete table of values for the graph of $y = 2 x (x - 1) (x - 3)$ .

x	0	0·2	0·5	0·8	1	1·5	2	2·2	2·5	2·8	3
y	0	0.9		0·7	0	−2.3		−4·2	−3·8	−2·0	0

(a) Calculate the missing values of $y$ .

(b) Using scales of 4 cm to 1 unit on the $x$ -axis and 2 cm to 1 unit on the $y$ -axis, draw the graph of $y = 2 x (x - 1) (x - 3)$ , for $0 \leq x \leq 3$ .

(c) Use your graph to find two values of $x$ which satisfy the equation $2 x (x - 1) (x - 3) = - 1$ .

(d) By drawing a suitable straight line on the same axes, use your graph to find the three values of $x$ which satisfy the equation $2 x (x - 1) (x - 3) = - x + 1$ .

11. The diagram in the answer space shows the straight line $y = 2 x + 1$ and part of the curve $y = \frac{12}{x}$ .

(a) Plot the three points on the curve $y = \frac{12}{x}$ for which $x = 6, 8$ and $12$ . Complete the curve for $4 \leq x \leq 12$ .

(b) Estimate, from the diagram,

(i) a solution of the equation $2 x + 1 = \frac{12}{x}$ ,

(ii) the gradient of the curve $y = \frac{12}{x}$ at the point (3, 4).

12. Answer the whole of this question on a sheet of graph paper. A building has vertical walls and stands on horizontal ground. A ball is thrown from a window of the building. At any instant the horizontal distance of the ball from the building is $x$ metres and the vertical height of the ball above the ground is $y$ metres. Some corresponding values of $x$ and $y$ are given in the following table.

x	0	1	2	3	4	5	6	7
y	20	24	26	26	24	20	14	6

(a) Using a scale of 2 cm to represent 1 metre on the $x$ -axis and 2 cm to represent 5 metres on the $y$ -axis, draw $x$ and $y$ axes for $0 \leq x \leq 8$ and $0 \leq y \leq 30$ . Plot the given values of $x$ and $y$ and draw a smooth curve through your points.

(b) Write down the height of the window above the ground.

(c) From the graph estimate

(i) the greatest height above the ground reached by the ball,

(ii) the horizontal distance from the building of the point where the ball strikes the ground,

(iii) the horizontal distance travelled by the ball whilst it is more than 23 metres above the ground.

(d) A wall of height 7 metres is situated $d$ metres from the building. Given that the ball passes 4 metres above the top of the wall, use your graph to estimate the value of $d$ .

13. Answer the whole of this question on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = 5 - \frac{x^{2}}{10} - \frac{4}{x}$ and some corresponding values are given in the following table.

x	0·5	0·7	1	2	3	4	5	6	7	8
y	−3·0	−0·8	0·9	2·6	2·8	2·4	1·7	0·7	−0·5	k

Calculate the value of $k$ .

Taking 2 cm to represent 1 unit on each axis, draw $x$ and $y$ axes for $0 \leq x \leq 8$ and $- 3 \leq y \leq 3$ .

Draw the graph of $y = 5 - \frac{x^{2}}{10} - \frac{4}{x}$ for the range $0 \cdot5 \leq x \leq 8$ .

(a) By drawing a tangent, find the gradient of the graph at the point where $x = 4$ .

(b) Using your graph, estimate

(i) the area between the graph, the $x$ -axis and the lines $x = 1$ and $x = 3$ ,

(ii) the solutions of the equation $6 - \frac{x^{2}}{10} - \frac{4}{x} = 0$ .

14. Answer the whole of this question on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = \frac{1}{2} (7 x - x^{2})$ . Some corresponding values are given in the following table.

x	0	1	2	3	3 $\frac{1}{2}$	4	5	6	7	8
y	0	3	5	6	a	6	5	3	0	b

(a) Calculate the value of $a$ and the value of $b$ .

(b) Taking 2 cm to represent 1 unit on each axis, draw the graph of $y = \frac{1}{2} (7 x - x^{2})$ for the range $0 \leq x \leq 8$ .

(c) By drawing the line $y = x + 1$ on your diagram, find the range of values of $x$ for which $\frac{1}{2} (7 x - x^{2}) > x + 1$ .

(d) Use your graph to estimate the two values of $x$ which satisfy the equation $7 x - x^{2} = 5$ .

(e) Estimate, from your graph, the area between the curve $y = \frac{1}{2} (7 x - x^{2})$ , the $x$ -axis and the lines $x = 1$ and $x = 4$ .

15. Answer the whole of this question on a sheet of graph paper.

(a) The diagram shows a rectangular sheet of cardboard measuring 17 cm by 8 cm. A square of side $x$ cm is cut out from each corner, and the cardboard is folded along the dotted lines to form an open rectangular box with base $A B C D$ and height $x$ cm. Show that the volume of the box is $2 x (17 - 2 x) (4 - x) cm3$ .

(b) Given that $y = 2 x (17 - 2 x) (4 - x)$ and that corresponding values of $x$ and $y$ are shown in the above table, calculate the value of $a$ and the value of $b$ .

x	0	0·5	1	1·5	2	2·5	3	3·5	4
y	0	56	90	105	104	90	66	a	b

(c) Using a scale of 2 cm to represent $\frac{1}{2}$ unit on the $x$ -axis and a scale of 2 cm to represent 20 units on the $y$ -axis, draw a graph of $y = 2 x (17 - 2 x) (4 - x)$ for values of $x$ in the range $0 \leq x \leq 4$ .

(d) Use your graph to estimate the volume of the box whose height is 2·3 cm.

(e) Use your graph to estimate the greatest possible volume of the box.

(f) Use your graph to estimate the volume of the box whose length is 15·2 cm.

16. Answer the whole of this question on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = \frac{x^{2}}{2} - 2 x$ and some corresponding values are given in the following table.

x	−1	0	0·5	1	2	3	4	5	6
y	2·5	0	−0·9	−1·5	−2	−1·5	0	2·5	a

(a) Calculate the value of $a$ .

(b) Taking 2 cm to represent 1 unit on each axis, draw the graph of $y = \frac{x^{2}}{2} - 2 x$ for values of $x$ in the range $- <1 \leq x \leq 6$ .

(c) Find, from the graph,

(i) the value of $y$ when $x = 4 ·7$ ,

(ii) the values of $x$ when $y = - 1$ .

(d) On the same axes, draw the graph of the straight line $y = 2 - \frac{x}{2}$ and use your graphs to solve the equation $\frac{x^{2}}{2} - 2 x = 2 - \frac{x}{2}$ .

(e) On your diagram, draw the line through the origin which is perpendicular to $y = 2 - \frac{x}{2}$ . Find the equation of this line.

17. Answer the whole of this equation on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = \frac{1}{5} (13 x - x^{2})$ . Some corresponding values of $x$ and $y$ are given in the following table.

x	0	2	4	6	8	10	12	14
y	0	4·4	7·2	8·4	8·0	6·0	2·4	a

(a) Calculate the value of $a$ .

(b) Taking 1 cm to represent 1 unit on each axis, draw the graph of $y = \frac{1}{5} (13 x - x^{2})$ for values of $x$ in the range $0 \leq x \leq 14$ .

(c) (i) Find the $x$ -coordinates of the points on the curve where $y = 4$ .

(ii) Write down and simplify the equation in $x$ which has these values as its solutions.

(d) Estimate the area between the curve $y = \frac{1}{5} (13 x - x^{2})$ , the $x$ -axis and the lines $x = 2$ and $x = 6$ .

(e) By drawing a tangent, find the gradient of the curve $y = \frac{1}{5} (13 x - x^{2})$ at the point (10, 6).

18. The diagram shows a sketch of part of the graph of $y = (x - 2) (x - 6)$ . The curve cuts the $x$ -axis at $P$ and $Q$ , and the $y$ -axis at $R$ .

(a) Write down the coordinates of $P$ , $Q$ and $R$ .

(b) The coordinates of the point $D$ are (1, $d$ ). Given that the point $D$ lies on the curve, calculate the value of $d$ .

(c) Given also that the straight line $y = x + k$ passes through $D$ , calculate the value of $k$ .

19. The diagram shows part of the graph of $y = x^{3} - 6 x^{2} + 8 x$ .

(a) Draw the tangent to the curve at the point (1, 3) and hence estimate the gradient of the curve at this point.

(b) On the diagram draw the line $y = 2 x - 1$ .

Estimate, from the diagram, a solution of the equation $x^{3} - 6 x^{2} + 8 x = 2 x - 1$ .

(c) Estimate, from the diagram, two solutions of the equation $x^{3} - 6 x^{2} + 8 x + 2 = 0$ .

20. The diagram shows part of the graph of $y = 8 - x - \frac{4}{x}$ .

(a) By drawing a tangent, estimate the gradient of the graph at the point (1, 5).

(b) By drawing another line on the graph, estimate the two solutions of the equation $8 - x - \frac{4}{x} = 2 x$ .

(c) Estimate the area between the graph, the $x$ -axis and the lines $x = 1$ , $x = 3$ .

21. Answer the whole of this question on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = \frac{1}{10} (120 - x^{2} - \frac{96}{x})$ . Some corresponding values of $x$ and $y$ are given in the table.

x	1	2	3	4	6	8	10	12
y	2·3	6·8	7·9	8·0	6·8	4·4	1·0	a

(a) Calculate the value of $a$ .

(b) Taking 1 cm to represent 1 unit on each axis, draw $x$ and $y$ axes for $0 \leq x \leq 12$ and $- 4 \leq y \leq 10$ . Draw the graph of $y = \frac{1}{10} (120 - x^{2} - \frac{96}{x})$ for values of $x$ in the range $1 \leq x \leq 12$ .

(c) (i) Use your graph to find the $x$ -coordinates of the points on the curve for which $y = 4$ ,

(ii) Write down, but do not simplify, an equation in $x$ which is satisfied by these values of $x$ .

(d) By drawing another line on your graph, find the solution of the equation $x = \frac{1}{10} (120 - x^{2} - \frac{96}{x})$ , which lies between $x = 4$ and $x = 8$ .

(e) By drawing a tangent, find the gradient of the curve $y = \frac{1}{10} (120 - x^{2} - \frac{96}{x})$ at the point (8, 4.4).

22. Answer the whole of this question on a sheet of graph paper.

The variables $x$ and $y$ are connected by the equation $y = \frac{x^{2}}{5} + \frac{12}{x} - 4$ . Some corresponding values, correct to one decimal place, are given in the following table.

x	1	1·5	2	3	4	5	6	7
y	8·2	4·5	2·8	t	2·2	3·4	5·2	7·5

(a) Calculate the value of $t$ .

(b) Taking 2 cm to represent 1 unit on each axis, draw the graph of $y = \frac{x^{2}}{5} + \frac{12}{x} - 4$ for values of $x$ in the range $1 \leq x \leq 7$ .

(c) Showing your method clearly, use your graph to find the solutions of the equation $\frac{x^{2}}{5} + \frac{12}{x} = 7$ in the range $1 \leq x \leq 7$ .

(d) (i) On the same axes draw the graph of the straight line $y = x$ .

(ii) Use your graphs to find the coordinates of a point on the curve $y = \frac{x^{2}}{5} + \frac{12}{x} - 4$ at which the gradient of the tangent is equal to 1.

23. Answer the whole of this question on a sheet of graph paper.

The variables $x$ and $y$ are connected by the equation $y = \frac{1}{10} x (15 - x^{2})$ . Some corresponding values are given in the following table.

x	−2	−1	0	1	2	2·5	3	4	4·5	5
y	−2·2	−1·4	0	1·4	2·2	2·2	1·8	−0·4	−2·4	p

(a) Calculate the value of $p$ .

(b) Taking 2 cm to represent 1 unit on each axis, draw the graph of $y = \frac{1}{10} x (15 - x^{2})$ for values of $x$ in the range $- 2 \leq x \leq 5$ .

(c) By drawing a tangent, find the gradient of the graph at the point where $x = 3 \cdot5$ .

(d) Using your graph, estimate

(i) the area, in square units, between the graph, the $x$ -axis and the lines $x = 1$ and $x = 3$ ,

(ii) two solutions of the equation $x (15 - x^{2}) = - 10$ .

24. On the axes in the answer space, sketch the graph of

(a) $y = 4$ ,

(b) $y = \frac{k}{x}$ , where $k$ is a positive constant.

25. Answer the whole of this question on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = x (4 - x^{2})$ . Corresponding values of $x$ and $y$ , corrected to 1 decimal place where necessary, are given in the following table.

x	−2·5	−2	−1·5	−1	−0·5	0	0·5	1	1·5	2	2·5
y	5·6	0	−2·6	−3	−1·9	0	1·9	3	2·6	0	−5·6

(a) Taking 2 cm to represent 1 unit on the $x$ -axis and 1 cm to represent 1 unit on the $y$ -axis, draw the graph of $y = x (4 - x^{2})$ for the range $- 2 \cdot5 \leq x \leq 2 \cdot5$ .

(b) Describe completely the symmetry of the curve you have drawn.

(c) From your graph, write down the range of values of $x$ for which the gradient of the curve is positive.

(d) Draw the line $y + x = 2$ on your diagram. Hence write down the three possible solutions to the equation $x (4 - x^{2}) = 2 - x$ .

(e) A second straight line cuts the curve in three points. Two of these points are (0, 0) and ( $h, k$ ).

Write down, in term of $h$ and/or $k$ , the coordinates of the third point.

26. Answer the whole of this question on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = x + \frac{6}{x}$ . Some corresponding values of $x$ and $y$ are given in the following table.

x	1	1·5	2	2·5	3	4	5	6	8
y	7	5.5	5	4.9	5	5.5	6.2	7	p

(a) Calculate the value of $p$ .

(b) Taking 2 cm to represent 1 unit on each axis, draw the graph of $y = x + \frac{6}{x}$ for values of $x$ in the range $1 \leq x \leq 8$ .

(c) Find, from your graph, the values of $x$ for which $y = 6$ .

(d) On the same graph draw the line $y = 7 - \frac{x}{2}$ , and use your graph to solve the equation $x + \frac{6}{x} = 7 - \frac{x}{2}$ .

(e) By drawing a suitable tangent to your curve, find the coordinates of the point at which the gradient of the tangent is equal to $- \frac{1}{2}$ .

27. Answer the whole of this question on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = \frac{1}{2} x^{2} (3 - x)$ . Some corresponding values, corrected to 1 decimal place where necessary, are given in the following table.

x

−1·5

−1·3

−1

−0·5

0

0·5

1

1·5

2

2·5

3

3·3

3·5

y

5·1

3·6

2

0·4

0

0·3

1

1·7

2

1·6

0

−1·6

−3·1

(a) Taking 2 cm to represent 1 unit on each axis, draw the graph of $y = \frac{1}{2} x^{2} (3 - x)$ for values of $x$ in the range $- 1 \cdot5 \leq x \leq 3 \cdot5$ .

(b) Use your graph to write down the value of the largest solution of the equation $\frac{1}{2} x^{2} (3 - x) = 1$ .

(c) By drawing a tangent, find the gradient of the curve at the point where $x = 2 \cdot5$ .

(d) Estimate the area, in square units, between the graph and the $x$ -axis for $0 \leq x \leq 3$ .

(e) Describe completely the symmetry of the graph.

28. The diagram shows the graph of $y = x^{2} - 4 x$ . The graph passes through the origin and crosses the $x$ -axis again at the point $A$ .

(a) Calculate the coordinates of $A$ .

(b) (i) Write down the equation of the line of symmetry of the graph,

(ii) Find the coordinates of the lowest point on the graph.

(c) There is a point on the graph, other than the point (0, 0), where the $x$ and $y$ coordinates are equal. Find the coordinates of this other point.

29. Answer the whole of this question on a sheet of graph paper. The variables $x$ and $y$ are connected by the equation $y = \frac{1}{5} (x^{2} + \frac{10}{x})$ . Some corresponding values of $x$ and $y$ corrected to 1 decimal place where necessary, are given in the table.

x	0·3	0·6	1	1·5	2	3	4	5	6
y	6.7	3.4	2.2	1.8	1.8	2.5	3.7	5.4	p

(a) Calculate the value of $p$ , giving your answer correct to 1 decimal place.

(b) Taking 2 cm to represent 1 unit on each axis, draw $x$ and $y$ axes for $0 \leq x \leq 6$ and $0 \leq y \leq 8$ . Draw the graph of $y = \frac{1}{5} (x^{2} + \frac{10}{x})$ for values of $x$ in the range $0 \cdot3 \leq x \leq 6$ .

(c) Use your graph to find the solutions of the equation $\frac{1}{5} (x^{2} + \frac{10}{x}) = 3$ in the range $0 \cdot3 \leq x \leq 6$ .

(d) By drawing a tangent, find the gradient of the curve $y = \frac{1}{5} (x^{2} + \frac{10}{x})$ at the point where $x = 4$ .

(e) (i) On the same axes, draw the graph of the straight line $2 y = x + 7$ .

(ii) Write down the $x$ coordinates of the points where the two graphs meet.

(iii) Write down, but do not simplify, an equation in $x$ which has these values as two of its solutions.

30. The diagram shows the graph of $y = 6 x - x^{2}$ . The graph passes through the origin and crosses the $x$ -axis again at the point $C$ .

(a) Calculate the coordinates of $C$ .

(b) (i) Write down the equation of the line of symmetry of the graph,

(ii) Find the coordinates of the highest point on the graph.

(c) There is a point on the graph, other than the point (0, 0), where the $x$ and $y$ coordinates are equal. Find the coordinates of this other point.

31. This table gives the $x$ and $y$ coordinates of some points which lie on a curve.

x	−2	−1	0	1	2	3	4	5
y	5	2	0	−1	−1	0	2	5

(a) Taking 2 cm to represent 1 unit on each axis, and values of $x$ and $y$ from −2 to 5, plot these points and draw a smooth curve through them.

(b) Write down the equation of the line of symmetry of this curve.

(c) The points (3·7, $k$ ) and ( $h$ , $k$ ) lie on the curve. Use your graph to find

(i) the value of $k$ ,

(ii) the value of $h$ .

(d) By drawing a tangent, find the gradient of the curve at the origin.

(e) The values of $x$ and $y$ are related by the equation $y = A x^{2} + B x$ .

(i) Use the fact that the point (4, 2) lies on the curve to show that $8 A + 2 B = 1$ .

(ii) Use another point on the curve to find a second equation connecting $A$ and $B$ . Hence calculate the value of $A$ and the value of $B$ .

32. A container is being filled with water flowing at a constant rate from a tap. At time $t$ seconds after the tap is turned on, the height of the water in the container is $h$ centimetres. Initially the container is empty.

(a) If the container is a cylinder, sketch the graph of $h$ against $t$ .

(b) If the container is a bottle, as shown below, sketch the graph of $h$ against $t$ .

33. On the axes in the answer space sketch the graphs of

(a) $x + y = 0$ ,

(b) $y = \frac{1}{x^{2}}$ .

34. The diagram shows the graphs of $y = x^{2} + x - 3$ and $y = x - 1$ for $- 3 \leq x \leq 2$ . The graphs intersect at $P$ and $Q$ . Use the diagram to answer the following questions. Do not redraw the graphs.

(a) Estimate the solutions of the equations

(i) $x^{2} + x - 3 = 0$ ,

(ii) $x^{2} + x = 5$ .

(b) (i) Write down the $x$ -coordinates of $P$ and $Q$ .

(ii) Write down, but do not simplify, the equation in $x$ which has these values as its solutions.

35.

The graph shows the relation between the number ( $n$ ) of units of electricity used and the total cost ( $C$ ) of an electricity bill.

(a) Use the graph to find

(i) the cost of the bill if 300 units are used,

(ii) the number of units used when the bill is $32·50.

(b) Given that the relation is $C = p n + q$ ,

(i) state the value of $q$ and explain its significance,

(ii) find the value of $p$ and explain its significance,

(iii) find the total cost of the bill if 1100 units are used.

36. Answer the whole of this question on a sheet of graph paper.

A stone is thrown from the top of a vertical cliff. Its position during its flight is represented by the equation $y = 56 + 10 x - x^{2}$ , where $y$ metres is the height of the stone above the sea and $x$ metres is its horizontal distance from the cliff.

(a) (i) Solve the equation $0 = 56 + 10 x - x^{2}$ ,

(ii) Explain briefly what the positive solution of this equation represents.

Some corresponding values of $x$ and $y$ are given in the following table.

x		0	2	4	6	8	10
y		56	72	80	80	72	56

(b) (i) By considering the symmetry of the $y$ values in the table, state the value of $x$ at which the stone reaches its greatest height.

(ii) Use this value of $x$ in the given equation to calculate the greatest height reached.

(c) Taking 2 cm to represent 1 metre on the $x$ -axis and 2 cm to represent 5 metres on the $y$ -axis, draw the graph of $y = 56 + 10 x - x^{2}$ for values of $x$ in the range $0 \leq x \leq 10$ and values of $y$ in the range $55 \leq y \leq 90$ .

(d) Use your graph to find how far the stone travels horizontally while its height is more than 76 metres.

37. Answer the whole of this question on a sheet of graph paper.

The volume of an open rectangular box, made of thin metal, is 7500 cm³.

The lengths of the edges of the base of the box are 30 cm and $x$ cm.

(a) Find, in terms of $x$ , an expression for

(i) the area of the base of the box,

(ii) the height of the box.

(b) The total external area, of the base and the four sides, is $A {cm}^{2}$ .

Show that $A = 500 + 30 x + \frac{15000}{x}$ .

(c) The table below shows some values of $x$ and the corresponding values of $A$ . The values of $A$ are given correct to the nearest integer, where appropriate.

x	10	12·5	15	20	25	30	35	40	45
A	2300	2075	1950	1850	1850	1900	1979	2075	2183

Using a scale of 2 cm to 5 units draw a horizontal $x$ axis for $10 \leq x \leq 45$ .

Using a scale of 2 cm to 100 units draw a vertical $A$ axis for $1800 \leq A \leq 2300$ .

Plot the points represented by the values in the table and join them with a smooth curve.

38.

The diagrams represent the cross sections, as seen from the side, of three rectangular swimming pools. All three pools have the same length and width. In Pools Y and Z the length of the shallow part is half the length of the pool. The pools are to be filled to a maximum depth of two metres. When full, Pools Y and Z are one metre deep at the shallow part. Water is pumped into all three pools at the same constant rate. The graph in the answer space shows the height of the water above the line $A B$ against time as one of the pools is being filled.

(a) Which pool is this?

(b) On the same axes draw the corresponding graphs for the other two pools. Label each graph clearly.

39. The diagram in the answer space shows the graphs of the curve $y = (x + 3) (5 - x)$ and the line $y = x - 4$ . The dotted line is the line of symmetry of the curve. The curve meets the $y$ -axis at $A$ , and the $x$ -axis at $B$ and $C$ as shown.

(a) Find the coordinates of $A$ .

(b) Find the coordinates of $B$ .

(c) Find the equation of the line of symmetry.

(d) On the diagram, shade the region defined by the inequalities $x \geq 0$ , $y \leq 0$ and $y \geq x - 4$ .

40. Answer the whole of this question on a sheet of graph paper.

The number of bacteria in a colony doubles every hour. The colony starts with 50 bacteria. The table below shows the number of bacteria in the colony after time $t$ .

Time in hours ( $t$ )	0	1	2	3	4	5	6	7
Number of bacteria ( $n$ )	50	100	200	400	800	1,600	3,200	6,400

(a) Using a horizontal scale of 2 cm to represent 1 hour and a vertical scale of 2 cm to represent 1000 bacteria, draw the graph of $n$ against $t$ .

(b) Use your graph to find the value of $n$ when $t = 6·5$ .

(c) (i) By drawing a tangent, find the gradient of the graph when $t = 5·5$ .

(ii) State briefly what this gradient represents.

(d) The number of bacteria in another colony is given by the equation $n = 4000 - 500 t$ .

(i) On the same axes, draw a graph to represent the number of bacteria in this colony.

(ii) Find the value of $t$ when the numbers in the colonies are equal.

(e) Given that the equation of the first graph is $n = k 2^{t}$ , find the value of $k$ .

41. In June 1995, 1 dollar = 3·56 Pula.

(a) On the axes in the answer space, draw a graph which you can use to convert from one currency to the other.

(b) Use your graph to estimate the cost in dollars of a T-shirt priced at 65·20 Pula.

42. Answer the whole of this question on a sheet of graph paper.

In the diagram, $A B C D$ represents a rectangular plot of area 40 square metres. $P Q R S$ represents the rectangular base of a shed positioned symmetrically on the plot. The distances, in metres, of the sides of the shed from the edges of the plot are as shown.

(a) Taking the length of $A B$ to be $x$ metres, write down expressions, in terms of $x$ , for the lengths

(i) $B C$ ,

(ii) $P Q$ ,

(iii) $Q R$ .

(b) Show that the area, $y$ square metres, of $P Q R S$ is given by $y = 48 - 2 x - \frac{160}{x}$ .

(c) The table below shows some values of $x$ and the corresponding values of $y$ . The values of $y$ are given correct to two decimal places where appropriate.

x (metres)	6	7	8	9	10	11	12	13
y (square metres)	9·33	11·14	12	12·22	12	11·45	10·67	9·69

Using a scale of 2 cm to 1 unit on each axis, draw $x$ and $y$ axes for $6 \leq x \leq 13$ and $9 \leq y \leq 13$ . On your axes plot the points given in the table and join them with a smooth curve.

(d) By using your graph, find

(i) the smaller value of $x$ for which the area of $P Q R S$ is 10 m²,

(ii) the value of $x$ for which the area of $P Q R S$ is maximum,

(iii) the dimensions of the base of the shed for which it has maximum area.

43. Answer the whole of this question on a sheet of graph paper.

(a) Table I below gives some values of $x$ and the corresponding values of $y$ , correct to two decimal places, where $y = x (1 + x) (3 - x)$ .

Table I
x	0	0·5	1	1·5	2	2·5	3
y	0	1·88	4	5·63	6	4·38	p

(i) Find the value of $p$ .

(ii) Using a scale of 4 cm to represent 1 unit draw a horizontal $x$ -axis for $0 \leq x \leq 3$ . Using a scale of 2 cm to represent 1 unit draw a vertical $y$ -axis for $0 \leq y \leq 8$ . On your axes plot the points given in the table and join them with a smooth curve.

(iii) Use your graph to find the greatest value of $x (1 + x) (3 - x)$ in the interval $0 \leq x \leq 3$ .

(b) Table II shows some corresponding values of $x$ and $y$ where $y = 2^{x}$ .

Table II
x	0	0·5	1	1·5	2	2·5	3
y	1	1·41	2	q	4	5·66	8

(i) Find the value of $q$ , correct to 2 decimal places.

(ii) On the axes used in part (a)(ii) draw the graph of $y = 2^{x}$ .

(c) From your graphs, find the values of $x$ in the interval $0 \leq x \leq 3$ , for which

(i) $x (1 +x)(3-x)=3$ ,

(ii) $x (1 + x) (3 - x) > 2^{x}$ .

44. Answer the whole of this question on a sheet of graph paper.

A toy firm makes a profit of $$ y$ when it produces $x$ toys, where $y$ is given by the formula $y = \frac{1}{10} (3 x - 20) (100 - x)$ for $0 \leq x \leq 80$ . The table below shows the profit which the firm makes when producing different numbers of toys.

Number of toys ( $x$ )	0	10	20	30	40	50	60	70	80
Profit ( $$ y$ )	−200	90	320	490	600	650	640	570	p

(a) Find the value of $p$ .

(b) Using a scale of 2 cm to represent 10 toys, draw a horizontal axis for $0 \leq x \leq 80$ . Using a scale of 2 cm to represent $100, draw a vertical axis for $- 200 \leq y \leq 700$ . On your axes plot the points given in the table and join them with a smooth curve.

(c) Use your graph to estimate

(i) the profit made when 25 toys are produced,

(ii) (a) the number of toys produced to give the maximum profit,

(b) the profit per toy when this number is produced,

(iii) the value of $y$ when $x = 5$ , explaining the significance of your answer.

(d) (i) On the same axes draw the line whose equation is $y = 9 x$ .

(ii) Hence find the two values of $x$ for which the profit is $9 per toy.

45. Answer the whole of this question on a sheet of graph paper.

When $x$ copies of a book are produced, the cost, $$ y$ , of each copy is given by the formula $y = 10 + \frac{2400}{x}$ .

(a) The table below gives some values of $x$ and the corresponding values of $y$ .

x	100	200	300	400	600	800	1200
y	34	22	18	16	14	13	p

(i) Find the value of $p$ .

(ii) Using a scale of 2 cm to represent 200 books, draw a horizontal $x$ -axis for $0 \leq x \leq 1200$ . Using a scale of 2 cm to represent $5, draw a vertical $y$ -axis for $0 \leq y \leq 40$ . On your axes plot the points given in the table and join them with a smooth curve.

(b) Use your graph to estimate the number of books to be printed if the cost of producing each book is $15.

(c) (i) By drawing a tangent, find the gradient of the curve at the point where $x = 300$ .

(ii) Describe briefly what this gradient represents.

(d) In order to sell $x$ books, the selling price of each book must be $(25 - \frac{x}{60})$ dollars.

(i) On the axes used in part (a), draw the graph of $y = 25 - \frac{x}{60}$ , for values of $x$ from 0 to 1200.

(ii) Use your graphs to find the range of the number of books that should be printed if no loss is to be made, assuming that all of the books will be sold.

(N98/2/8)

46. Answer the whole of this question on a sheet of graph paper.

A particle was projected directly up a slope. Its distance, $d$ metres, from the bottom of the slope, $t$ seconds after it was projected, is given in the table below.

t	0	0·5	1	2	3	4	5	6	6·5	7
d	0	1·58	3·02	5·48	7·38	8·72	9·5	9·72	9·72	9·72

(a) Using a horizontal scale of 2 cm to represent 1 second, and a vertical scale of 1 cm to represent 1 metre, draw a graph of $d$ against $t$ .

(b) Use your graph to find the distance of the particle from the bottom of the slope when $t = 2·5$ .

(c) What happened to the particle after approximately 6 seconds?

(d) (i) By drawing a tangent, find the gradient of your curve when $t = 4$ .

(ii) State briefly what this gradient represents.

(e) Calculate the average speed of the particle during the first 6 seconds.

(f) At the instant the particle was projected, another particle was projected down the slope from a point 10 metres from the bottom. This particle moved directly down the slope at a constant speed of 2 m/s.

(i) On the same axes, draw the graph which shows the position of this particle.

(ii) Use your graphs to find when the particles passed each other.

47. The diagram shows the line $y = m x$ .

On the same diagram, sketch and label

(a) $y = m x + 2$ ,

(b) $y = - m x$ .

48. Answer the whole of this question on a sheet of graph paper.

The variables $x$ and $y$ are connected by the equation $y = \frac{x^{2}}{6} + \frac{12}{x} - 6$ .

The table below shows some corresponding values of $x$ and $y$ . The values of $y$ are given correct to one decimal place where appropriate.

x	1	1·5	2	3	4	5	6	7
y	6·2	2·4	0·7	−0·5	−0·3	0·6	2	k

(a) Calculate the value of $k$ , correct to one decimal place.

(b) Using a scale of 2 cm to 1 unit on each axis, draw a horizontal $x$ -axis for $0 \leq x \leq 8$ and a vertical $y$ -axis for $- 1 \leq y \leq 7$ . On your axes, plot the points given in the table and join them with a smooth curve.

(c) By drawing a tangent, find the gradient of the curve at the point (1·5, 2·4).

(d) Showing your method clearly, use your graph to find the values of $x$ in the range $1 \leq x \leq 7$ for which $\frac{x^{2}}{6} + \frac{12}{x} = 7$ .

(e) (i) On the same axes, draw the graph of the straight line $y = \frac{x}{4}$ .

(ii) Using your graphs, find the values of $x$ in the range $1 \leq x \leq 7$ for which $\frac{x^{2}}{6} + \frac{12}{x} - 6 \leq \frac{x}{4}$ .

49. The diagram shows the line $y = m x$ .

On the same diagram, sketch, and label clearly the lines

(a) $y = 3 m x$ ,

(b) $y = m x - 4$ .

50. Answer the whole of this question on a sheet of graph paper.

The table below gives some values of $x$ and the corresponding values of $y$ , given correct to two decimal places, for $y = \frac{1}{4} (x^{3} - 6 x^{2} + 8 x)$ .

x

−1

−0·5

0

0·5

1

1·5

2

2·5

3

3·5

4

4·5

5

y

−3·75

−1·41

0

0·66

0·75

0·47

0

−0·47

−0·75

−0·66

0

1·41

3·75

(a) Using a scale of 2 cm to represent 1 unit on each axis, draw a horizontal $x$ -axis for $- 1 \leq x \leq 5$ and a vertical $y$ -axis for $- 4 \leq y \leq 4$ . On your axes, plot the points given in the table and join them with a smooth curve.

(b) Describe the symmetry of this curve.

(c) Use your graph to solve the equations

(i) $\frac{1}{4} (x^{3} - 6 x^{2} + 8 x) = - 1$ ,

(ii) $x^{3} - 6 x^{2} + 8 x = 8$ .

(d) By drawing a tangent, find the gradient of the curve at the origin.

(e) The line $y = m x$ intersects the curve $y = \frac{1}{4} (x^{3} - 6 x+8x)$ at three points. Find the least possible value of $m$ .

51. You are asked to investigate the price that a shop should charge for a particular toy. When the toy is priced at $$ x$ , the shop sells $(90 - 10 x)$ toys.

(a) Write down an expression, in terms of $x$ , for the total amount, in dollars, received by the shop for the sale of these toys.

(b) Find the value of $x$ if the amount received is to be as large as possible. You should use the grid and/or the table to help you in your investigation. Marks will be awarded for clear working.

x
1
2
•
•
•

52. Answer the whole of this question on a sheet of graph paper.

The table below gives some values of $x$ and the corresponding values of $y$ , correct to two decimal places, where $y = x (1 + x) (4 - x)$ .

x	−1	−0·5	0	0·5	1	1·5	2	2·5	3	3·5	4
y	0	−1·13	0	2·63	6	9·38	p	13·13	12	7·88	q

(a) (i) Find the value of $p$ and the value of $q$ .

(ii) Using a scale of 2 cm to 1 unit, draw a horizontal $x$ -axis for $- 1 \leq x \leq 4$ .

Using a scale of 1 cm to 1 unit, draw a vertical $y$ -axis for $- 2 \leq y \leq 14$ .

On your axes, plot the points given in the table and join them with a smooth curve.

(iii) Using your graph, find the values of $x$ for which $y = 3$ .

(b) By drawing a tangent, find the gradient of the curve at the point where $x = 3$ .

(c) On the axes used in part (a), draw the graph of $y = 10 - x$ for values of $x$ in the range $- 1 \leq x \leq 4$ .

(d) Write down, and simplify, the cubic equation which is satisfied by the values of $x$ at the points where the two graphs intersect.

53. You are asked to investigate the area of a rectangle with width $x$ and length $70 - 10 x$ .

(a) Write down an expression, in terms of $x$ , for the area of the rectangle.

(b) Find the value of $x$ if the area is to be as large as possible. You should use the grid and/or the table to help you in your investigation. Marks will be awarded for clear working.

x
1
2
•
•
•

54. The diagram in the answer space shows the graph of $y = 2^{x}$ .

(a) By drawing a tangent, find an estimate of the gradient of this curve at $x = 2$ .

(b) Use the graph to find the value of

(i) $2^{2 \cdot 5}$ ,

(ii) $n$ if $2^{n} = 5$ .

(J2001/1/14)

55. Answer the whole of this question on a sheet of graph paper.

The diagram shows a sketch of the curve $y = 36 - x^{2}$ . The curve crosses the axes at $H$ , $K$ and $L$ . $O$ is the origin.

(a) Show that the area of triangle $O H K$ is 108 square units.

The point $P$ lies on the curve between $H$ and $K$ . The coordinates of $P$ are ( $x, y$ ).

$P Q R S$ is a rectangle with $Q$ on the curve and $R$ and $S$ on the $x$ -axis.

The area of the rectangle $P Q R S$ is $A$ square units.

(b) Show that $A = 72 x - 2 x^{3}$ .

(c) The table below gives some values of $x$ and the corresponding values of $A$ , where $A = 72 x - 2 x^{3}$ .

x	0	1	2	3	4	5	6
A	0	70	128	t	160	110	0

(i) Find the value of $t$ .

(ii) Using a scale of 2 cm to represent 1 unit, draw a horizontal $x$ -axis for $0 \leq x \leq 6$ .

Using a scale of 1 cm to represent 20 units, draw a vertical $A$ -axis for $0 \leq A \leq 180$ .

On your axes, plot the points given in the table and join them with a smooth curve.

(iii) Use your graph to find the two values of $x$ when $A = 108$ .

(iv) Find the lengths of the sides of the rectangle $P Q R S$ when its area is equal to the area of triangle $O H K$ and $P Q$ is less than 4 units.

56. The graph of $y = (x + 2) (3 - x)$ cuts the $x$ axis at $A$ and $B$ . It cuts the $y$ axis at $C$ . Find

(a) the coordinates of the point $C$ ,

(b) the length $A B$ ,

(c) the equation of the straight line $A C$ .

57. The point (1,1) is marked on the diagram.

On the diagram, sketch the graph of $y = \frac{1}{x}$ .

58. Answer the whole of this question on a sheet of graph paper.

The table gives some values of $x$ and the corresponding values of $y$ , correct to one decimal place, where $y = 4 x + \frac{25}{x^{2}}$ .

x	1	1·25	1·5	2·0	2·5	3·0	3·5	4·0
y	p	21·0	17·1	14·3	14·0	14·8	16·0	17·6

(a) Find the value of $p$ .

(b) Using a scale of 4 cm to represent 1 unit, draw a horizontal $x$ -axis for $0 \leq x \leq 4$ .

Using a scale of 4 cm to represent 10 units, draw a vertical $y$ -axis for $0 \leq y \leq 30$ .

On your axes, plot the points given in the table and join them with a smooth curve.

(c) Use your graph to find

(i) a solution of $4 x + \frac{25}{x^{2}} = 20$ ,

(ii) the least value of $4 x + \frac{25}{x^{2}}$ .

(d) By drawing a tangent, find the gradient of the curve at the point where $x = 1·5$ .

(e) On the axes used in part (b), draw the graph of the straight line $y = 25 - 3 x$ for values from $x = 0$ to $x = 4$ .

(f) (i) Write down the $x$ coordinates of the points at which the two graphs intersect.

(ii) Find the equation, in the form $a x^{3} + b x^{2} + c x + d = 0$ , which is satisfied by the values of $x$ found in part (f)(i).

Unit 3E

Algebra and Graphs — Graphs