Figure 1 shows a sketch of the curve with equation $y = f(x)$. The curve passes through the origin O and through the point (6, 0). The maximum point on the curve is (3, 5). On separate diagrams, sketch the curve with equation a) $y = 3f(x)$. b) $y = f(x + 2)$. On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the x-axis.

A-Level Edexcel Maths Pure Inequalities: Figure 1 shows a sketch of the c

Figure 1 shows a sketch of the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 6 - 2005 - Paper 1

Question 6

Figure 1 shows a sketch of the curve with equation $y = f(x)$. The curve passes through the origin O and through the point (6, 0). The maximum point on the curve is ... 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 6 - 2005 - Paper 1

Step 1

a) $y = 3f(x)$

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Answer

To sketch the curve $y = 3f(x)$ :

Stretch Vertically: The graph of $f(x)$ is stretched vertically by a factor of 3. Therefore, the maximum point (3, 5) will move to (3, 15).
Points of Intersection: The curve still passes through the origin (0, 0) and will also cross the x-axis at the point (6, 0) just as the original function does.
Drawing the Curve: Sketch the curve starting from (0, 0) up to (3, 15), then it will decline towards the x-axis crossing at (6, 0). Ensure that the shape of the curve maintains its properties with a rounded top.

The maximum point is (3, 15) and it crosses the x-axis at (0, 0) and (6, 0).

Step 2

b) $y = f(x + 2)$

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Answer

To sketch the curve $y = f(x + 2)$ :

Horizontal Shift: The function $f(x)$ is shifted horizontally to the left by 2 units. The maximum point (3, 5) will now be at (1, 5).
Points of Intersection: The curve will now cross the x-axis at new points, which can be found from the original intersections. Since the original intersections were at (0, 0) and (6, 0), they will now shift to (-2, 0) and (4, 0).
Drawing the Curve: Sketch the curve starting from (-2, 0) rising up to (1, 5) and then descending to (4, 0). Ensure the shape remains rounded and smoothly connects the points.

The maximum point is (1, 5) and it crosses the x-axis at (-2, 0) and (4, 0).

Figure 1 shows a sketch of the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 6 - 2005 - 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 6 - 2005 - Paper 1

a) $y = 3f(x)$

b) $y = f(x + 2)$

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