Figure 1 shows a sketch of the curve with equation $y = f(x)$. The curve has a maximum point A at $(-2, 3)$ and a minimum point B at $(3, -5)$. On separate diagrams sketch the curve with equation: a) $y = f(x + 3)$ b) $y = 2f(x)$ On each diagram show clearly the coordinates of the maximum and minimum points. The graph of $y = f(x) + a$ has a minimum at $(3, 0)$, where $a$ is a constant. c) Write down the value of $a$.

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Figure 1 shows a sketch of the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 1

Question 8

Figure 1 shows a sketch of the curve with equation $y = f(x)$. The curve has a maximum point A at $(-2, 3)$ and a minimum point B at $(3, -5)$. On separate diagram... 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 - 2010 - Paper 1

Step 1

a) $y = f(x + 3)$

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Answer

To sketch the curve for the equation $y = f(x + 3)$ , we perform a horizontal translation to the left by 3 units. This shifts the coordinates of the points A and B:

Point A which was at $(-2, 3)$ moves to $(-5, 3)$ .
Point B which was at $(3, -5)$ moves to $(0, -5)$ .

The maximum point A will still be at $(-5, 3)$ and the minimum point B will be located at $(0, -5)$ . Ensure that the shape of the curve remains intact, with the correct maximum and minimum points clearly identified.

Step 2

b) $y = 2f(x)$

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Answer

For the equation $y = 2f(x)$ , we scale the function vertically by a factor of 2. This implies that the y-coordinates of the points will be doubled:

The maximum point A, initially at $(-2, 3)$ , will now be at $(-2, 6)$ as $3 imes 2 = 6$ .
The minimum point B, originally at $(3, -5)$ , will now be at $(3, -10)$ as $-5 imes 2 = -10$ .

The graph will show A at $(-2, 6)$ and B at $(3, -10)$ , while ensuring the shape of the curve preserves the behavior between the maximum and minimum.

Step 3

c) Write down the value of a.

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Answer

From the information given, the graph of $y = f(x) + a$ has a minimum at $(3, 0)$ . Thus:

Since point B was originally at $(3, -5)$ in the function $f(x)$ , we have $f(3) = -5$ .

So, for $f(x) + a = 0$ at $x = 3$ : $-5 + a = 0$ This leads to: $a = 5.$ Therefore, the value of $a$ is 5.

Figure 1 shows a sketch of the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 1

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 1

a) $y = f(x + 3)$

b) $y = 2f(x)$

c) Write down the value of a.

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