The diagram shows part of the graph of y = x^2 - 2x + 3. (a) By drawing a suitable straight line, use your graph to find estimates for the solutions of x^2 - 3x - 1 = 0. (b) P is the point on the graph of y = x^2 - 2x + 3 where x = 2. Calculate an estimate for the gradient of the graph at the point P.

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The diagram shows part of the graph of y = x^2 - 2x + 3 - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 2

Question 20

The diagram shows part of the graph of y = x^2 - 2x + 3. (a) By drawing a suitable straight line, use your graph to find estimates for the solutions of x^2 - 3... show full transcript

Worked Solution & Example Answer:The diagram shows part of the graph of y = x^2 - 2x + 3 - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 2

Step 1

By drawing a suitable straight line, use your graph to find estimates for the solutions of x^2 - 3x - 1 = 0.

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Answer

To solve the equation $x^2 - 3x - 1 = 0$ , we rearrange this as $y = x^2 - 3x - 1$ and plot it on the same graph as $y = x^2 - 2x + 3$ . By plotting the two curves, we look for the intersection points. I would recommend drawing the straight line $y = 0$ and identifying points where the two curves intersect. From the graph, the estimated solutions are approximately $x hickapprox -0.3$ and $x hickapprox 3.2$ .

Step 2

Calculate an estimate for the gradient of the graph at the point P.

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Answer

To estimate the gradient at the point P where $x = 2$ , we need to find the derivative of $y = x^2 - 2x + 3$ . The derivative is given by:
$rac{dy}{dx} = 2x - 2.$
Substituting $x = 2$ , we find:
$rac{dy}{dx}|_{x=2} = 2(2) - 2 = 2.$
Thus, the estimated gradient of the graph at point P is approximately $2$ .

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