Practice

Functions and Graphs

1. Express each of the following equations in the form a(x + b)2 + c. Hence, sketch each of the following graphs, stating clearly the coordinates of the points of the x-intercept and y-intercept (if any) and the coordinates of the maximum or minimum turning point. Draw also its axis of symmetry and write down the equation of the axis of symmetry.

(a) y = x2 – 4x + 7

(b) y = 3x2 + 3x – 2

(c) y = (x + 3)(x + 6)

(d) y = 3 – 15x – 3x2

2. If the graph of y = x2 – 10x + 2 is drawn, what is the equation of the straight line graph suitable to be added into the graph of y = x2 – 10x + 2 in order to solve the equation x2 + 14x + 60 = 0?

3. The curve is a graph of the function y = 2x. The points (2, a) and (b, 8) lie on the curve.

Find the value of a and of b.

4. The two curves are the branches of the graph of y = 1/x. The straight line y = 2x + 2 cuts the curves at points A and B. Find the coordinates of point A and of point B.

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• (2, a)

• (b, 8)

y = 2x + 2

y = 1/x

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

The table below shows the corresponding values of x and y for the equation y = x2.

x –5 –4 –3 –2 –1 0 1 2 3 4 5y 25 16 a 4 1 0 b 4 9 c 25

(a) Find the value of a, of b and of c.

(b) Using a scale of 1 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 = x2 for –5 ≤ x ≤ 5.

(c) By constructing a tangent at x = 2, estimate the gradient of the curve when x = 2.

(d) By drawing suitable straight line graphs on the same axes, solve the following equations.

(i) x2 = 2x + 10

(ii) x = ½x2 – 7

(iii) 6 – 3x – x2 = 0

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

The table below shows the corresponding values of x and y for the equationy = 18x – 30 – x3, corrected to 1 decimal place.

x 0 0.5 1 1.5 2 2.5 3 3.5 4y –30 a –13 b –2 c –3 –9.9 –22

(a) Find the value of a, of b and of c.

(b) Using a scale of 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 = 18x – 30 – x3 for 0 ≤ x ≤ 4.

(c) Use your graph to find and write down the coordinates of the turning point.

(d) Find the gradient of the curve at the point where x = 3.5 by drawing a tangent.

(e) The table below shows the corresponding values of x and y for the equation y = –x3, corrected to 1 decimal place.

x 0 0.5 1 1.5 2 2.5y 0 a –1 b –8 c

(i) Find the value of a, of b and of c.

(ii) On the same pair of axes, draw the graph of y = –x3 and find the point of intersection of the two curves. Verify your answer by calculation.

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7. On these diagrams, sketch the graphs of the given equations.

y = x2

y = x2 – 2 y = 2x

y = –2x

End of Practice

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