Figure 1 shows a sketch of the curve with equation $y = f(x)$. The curve passes through the points $(0, 3)$ and $(4, 0)$ and touches the $x$-axis at the point $(1, 0)$. On separate diagrams, sketch the curve with equation (a) $y = f(x + 1)$. (b) $y = 2f(x)$. (c) $y = f\left( \frac{1}{2}x \right)$. On each diagram show clearly the coordinates of all the points at which the curve meets the axes.

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Figure 1 shows a sketch of the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 1

Question 8

Figure 1 shows a sketch of the curve with equation $y = f(x)$. The curve passes through the points $(0, 3)$ and $(4, 0)$ and touches the $x$-axis at the point $(1, 0... 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 - 2006 - Paper 1

Step 1

a) $y = f(x + 1)$

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Answer

To sketch the diagram for $y = f(x + 1)$ , we need to shift the original curve left by 1 unit. The points transform as follows:

From $(0, 3)$ to $(-1, 3)$
From $(4, 0)$ to $(3, 0)$
The point of tangency at $(1, 0)$ shifts to $(0, 0)$ .

The new curve should pass through the points $(-1, 3)$ , $(0, 0)$ , and $(3, 0)$ . It remains a similar shape but shifted accordingly.

Step 2

b) $y = 2f(x)$

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Answer

To sketch the diagram for $y = 2f(x)$ , we need to stretch the original curve vertically by a factor of 2. The points transform as follows:

The point $(0, 3)$ becomes $(0, 6)$
The point $(4, 0)$ stays at $(4, 0)$
The point of tangency at $(1, 0)$ stays at $(1, 0)$ .

The new curve should pass through $(0, 6)$ and $(4, 0)$ , maintaining the shape but with increased height.

Step 3

c) $y = f\left( \frac{1}{2}x \right)$

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Answer

To sketch the diagram for $y = f\left( \frac{1}{2}x \right)$ , we need to stretch the original curve horizontally by a factor of 2. The points transform as follows:

From $(0, 3)$ to $(0, 3)$ (no change)
From $(4, 0)$ to $(8, 0)$
The point of tangency at $(1, 0)$ stretches to $(2, 0)$ .

The new curve should pass through $(0, 3)$ , $(2, 0)$ , and $(8, 0)$ , reflecting the horizontal stretching.

Figure 1 shows a sketch of the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 8 - 2006 - 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 - 2006 - Paper 1

a) $y = f(x + 1)$

b) $y = 2f(x)$

c) $y = f\left( \frac{1}{2}x \right)$

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