Figure 1 shows a sketch of the curve with equation $y = f(x)$. The curve crosses the x-axis at the points (1, 0) and (4, 0). The maximum point on the curve is (2, 5). In separate diagrams sketch the curves with the following equations. On each diagram show clearly the coordinates of the maximum point and of each point at which the curve crosses the x-axis. a) $y = 2f(x)$. b) $y = f(-x)$. c) The maximum point on the curve with equation $y = f(x + a)$ is on the y-axis. Write down the value of the constant $a$.

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Figure 1 shows a sketch of the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 8 - 2008 - Paper 2

Question 8

Figure 1 shows a sketch of the curve with equation $y = f(x)$. The curve crosses the x-axis at the points (1, 0) and (4, 0). The maximum point on the curve is (2, 5)... show full transcript

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 8 - 2008 - Paper 2

Step 1

a) $y = 2f(x)$

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Answer

To sketch the curve $y = 2f(x)$ , we need to stretch the original curve vertically by a factor of 2. The x-intercepts remain the same, so they are (1, 0) and (4, 0). The maximum point, originally at (2, 5), is scaled to (2, 10). Thus, the new curve has the same shape as the original but reaches a maximum of 10 at $x = 2$ .

Step 2

b) $y = f(-x)$

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Answer

This transformation reflects the curve across the y-axis. The x-intercepts will now be at (-1, 0) and (-4, 0). The maximum point (2, 5) will reflect to (-2, 5). The new curve will maintain the same shape as the original but reversed horizontally, indicating that the peak is now on the left side of the y-axis.

Step 3

c) Write down the value of the constant $a$.

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Answer

For the maximum point of the curve $y = f(x + a)$ to lie on the y-axis, the x-coordinate of the maximum must be 0. Given the maximum of the original curve is at $x = 2$ , we need to find $a$ such that:

$2 + a = 0$

Solving for $a$ gives:

$a = -2.$

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