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Here is a sketch of a curve - Edexcel - GCSE Maths - Question 18 - 2018 - Paper 1

Question 18

Here is a sketch of a curve. The equation of the curve is $y = x^2 + ax + b$ where $a$ and $b$ are integers. The points $(0, -5)$ and $(5, 0)$ lie on the curve. F... show full transcript

Worked Solution & Example Answer:Here is a sketch of a curve - Edexcel - GCSE Maths - Question 18 - 2018 - Paper 1

Step 1

Substituting the points into the curve equation

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Answer

We start by substituting the given points into the equation of the curve. For the point $(0, -5)$ :

$-5 = (0)^2 + a(0) + b \\ -5 = b$

This gives us: $b = -5$ .

Next, we substitute the point $(5, 0)$ :

$0 = (5)^2 + a(5) + b \\ 0 = 25 + 5a - 5 \\ 0 = 5a + 20$

Rearranging gives us: $5a = -20$ , thus $a = -4$ .

Step 2

Finding the coordinates of the turning point

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Answer

The equation of the curve is now:

$y = x^2 - 4x - 5$

To find the turning point, we complete the square:

$y = (x^2 - 4x) - 5 \\ = (x^2 - 4x + 4) - 4 - 5 \\ = (x - 2)^2 - 9$

This shows that the vertex (turning point) is at $(2, -9)$ .

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