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Sketching Graphs and Exponential Graphs

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 12 marks

The curve y = x^2 + px + q crosses the x-axis at (-2,\ 0) and (6,\ 0)

Work out the coordinates of the turning point of the curve.

Write your answer as (x, y)

Hint

A parabola is symmetrical, so its turning point is exactly halfway between the two roots.

Worked solution
  1. The turning point is halfway between the roots: x=\dfrac{-2+6}{2}=2
  2. The roots -2 and 6 give y=(x+2)(x-6)
  3. At x=2: y=(2+2)(2-6)=4\times(-4)=-16
  4. Turning point (2,\ -16)

Question 22 marks

The graph of y = a \times b^{x} passes through the points (0,\ 4) and (1,\ 6)

Work out the value of y when x = -1

Give your answer as a fraction in its simplest form.

Hint

Substitute x=0 first: b^0=1, so this point tells you a straight away.

Worked solution
  1. At (0,\ 4): b^0=1, so a=4
  2. At (1,\ 6): 4\times b=6, so b=\dfrac32
  3. At x=-1: y=4\times\left(\dfrac32\right)^{-1}=4\times\dfrac23
  4. y=\dfrac83

Question 3Challenge4 marks

The grid shows the graph of y = x^2 + 3x - 1 for values of x from -5 to 2

The equation x(x + 4) = 3 can be solved by drawing a suitable straight line on the grid.

(a)

Work out the equation of the straight line that should be drawn.

Give your answer in the form y = mx + c

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

When your answer to part (a) is correct, the line is drawn on the grid.

Use the graph to solve x(x + 4) = 3

Give your answers to 1 decimal place.

2 marks

Give every value, separated by commas

Hint

Rearrange the equation so that one side is exactly x^2+3x-1; the other side is the line to draw. Answer part (a) before attempting part (b).

Worked solution

Part (a)

  1. Expand the bracket: x^2+4x=3
  2. So x^2+4x-3=0
  3. Write the left-hand side as (x^2+3x-1)+x-2
  4. So x^2+3x-1+x-2=0, which gives x^2+3x-1=-x+2
  5. The line to draw is y=-x+2

Part (b)

  1. The line y=-x+2 goes through (0,\ 2) and (2,\ 0)
  2. It crosses the curve twice
  3. Read the x-value at each crossing
  4. x=0.6 and x=-4.6

More on this topic: Sketching Graphs and Exponential Graphs worksheet with full solutions

All AQA Level 2 Further Maths practice questions

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