Algebra worksheet — GCSE Higher

1.The graph of y = f(x) passes through the point (0, 4). Work out the y-coordinate of the point where the graph of y = −f(x) + 5 crosses the y-axis.

2.f(x) = x + 3 and g(x) = 2x. Work out fg(x).

3.Make x the subject of the formula y = 4(x − 6).

4.Solve the inequality 2(3x − 1) ≥ 4x + 8.

5.Which of these equations is equivalent to 5x − 2 = 3x + 8?

6.The 2nd term of a geometric sequence is 6 and the 4th term is 24. Given that all the terms of the sequence are positive, work out the common ratio.

7.Work out the maximum value of y = sin x, and the smallest positive value of x, in degrees, at which it occurs.

8.A company's weekly profit, P thousand pounds, when it makes x thousand items is modelled by P = x² − 8x + 12, for x ≥ 0. Work out the values of x for which the company makes a loss.

9.A straight line has equation y = 4x + 3. A second line is parallel to the first line and passes through the point (0, −5). Work out the equation of the second line.

10.The point (9, 12) lies on the circle x² + y² = 225, which has centre (0, 0). The tangent to the circle at (9, 12) crosses the x-axis at the point P. Work out the x-coordinate of P.

11.A graph passes through the point (0, 1). As x increases through positive values, the graph rises more and more steeply, and as x decreases through negative values, the graph gets closer and closer to the x-axis without ever reaching it. Which of these could be the equation of the graph?

12.A square has vertices at (−2, 3), (−2, −3) and (4, −3). Write down the coordinates of the fourth vertex.

13.Which of these statements is an equation with exactly one solution?

14.A geometric sequence has first term 4 and common ratio √2. Work out the sum of the first three terms of the sequence, giving your answer in the form a + b√2.

15.A rectangular sheep pen has perimeter P, length l and width w, connected by the formula P = 2l + 2w. A farmer has 60 m of fencing for the perimeter and sets the width at 8 m. Work out the length of the pen.