Algebra worksheet — GCSE Higher

1.A sequence has the position-to-term rule n² + 2, where n is the position number. Work out the 6th term.

2.A quadratic graph has equation y = (x − 4)². A student says this graph crosses the x-axis at two different points. Explain why the student is wrong.

3.A car park charges a £4 fixed fee plus £3 for each hour. Kofi has exactly £25 to spend on parking. Using the inequality 4 + 3h ≤ 25, work out the greatest number of whole hours, h, he can park for.

4.The graph of y = f(x) has x-intercepts at x = −2 and x = 6 and crosses the y-axis at (0, −12). Work out the x-intercepts and the y-intercept of y = −f(x).

5.A tap fills a tank at a varying rate. A graph shows the rate of flow, in litres per minute, against time, in minutes. What does the area under this graph represent?

6.A number machine multiplies its input by 3 and then adds 7. The output is 1. Work out the input.

7.The nth term of a sequence is 4n − 3. Work out the 7th term of the sequence.

8.A gym charges a joining fee plus a monthly fee. Anna paid £100 in total after 3 months of membership. Ben paid £160 in total after 6 months of membership (same joining fee and monthly fee as Anna). Work out the monthly fee.

9.Work out the integer values of x that satisfy x² − 2x − 8 ≤ 0.

10.A number machine multiplies its input by 2 and then subtracts 5. Work out the output when the input is 6.

11.How many turning points does the graph of y = x² − 7x + 10 have?

12.f(x) = (x + 1)/2. Find f⁻¹(x).

13.A circle has centre O(0, 0) and equation x² + y² = 169. The point Q has coordinates (10, 11). Work out which of these gives the correct position of Q together with correct working.

14.Factorise fully 5x + 5y − 5

15.A runner's journey is shown on a distance-time graph: she runs 8 km in the first hour, rests for 30 minutes with no distance gained, then runs a further 4 km in the next 30 minutes. Work out her average speed for the whole journey, in kilometres per hour.