Algebra worksheet — GCSE Higher

1.A rule turns each input x into an output y. An input of 1 gives an output of 1, an input of 2 gives an output of 4 and an input of 3 gives an output of 9. Work out the rule.

2.Describe a sequence of two transformations that maps the graph of y = x² onto the graph of y = −(x − 5)².

3.A circle has equation x² + y² = 25. Does the point (3, 4) lie on this circle?

4.A water tank is a cuboid with a square base of side x metres and height (x + 1) metres. Its volume is 10 m³. This gives x³ + x² − 10 = 0, which can be solved using the iterative formula xₙ₊₁ = ∛(10 − xₙ²). Taking x₀ = 2, so that x₁ is the value found after the formula has been used once, work out x₃ correct to 3 decimal places.

5.By completing the square, find the turning point of the curve y = x² + 8x − 3.

6.A car's speed increases steadily while it accelerates. Its speed v m/s after t seconds is given by v = u + at, where u is the starting speed in m/s and a is the acceleration in m/s². A car starts at u = 4 m/s and accelerates at a = 2 m/s² until it reaches v = 20 m/s. Work out t, the time taken in seconds.

7.A machine bought for £2,400 loses 12% of its value every year. After how many complete years does its value first fall below £1,500?

8.Write down the gradient of the line with equation y = −x + 3.

9.The equation 3(2x − 1) = 4x + 9 is rearranged by expanding the brackets. Which of these is the correctly expanded equation?

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

11.Solve the inequality x² ≥ 16, giving your answer using set notation.

12.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?

13.A van's value depreciates by 18% each year. After 2 years it is worth £5,379.20. Work out its value when it was new, correct to the nearest pound.

14.Solve 2x² − 32 = 0.

15.a = 2 and b = 5. Work out the value of a²b.