Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x^2(9 - 2x) There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A. (a) Write down the coordinates of the point A. (b) On separate diagrams sketch the curve with equation (i) y = f(x + 3) (ii) y = f(3x) On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. (c) The curve with equation y = f(x) + k, where k is a constant, has a maximum point at (3, 10). (c) Write down the value of k.

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

Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x^2(9 - 2x) There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A - Edexcel - A-Level Maths Pure - Question 2 - 2011 - Paper 1

Question 2

Figure-1-shows-a-sketch-of-the-curve-C-with-equation-y-=-f(x)-where-f(x)-=-x^2(9---2x)--There-is-a-minimum-at-the-origin,-a-maximum-at-the-point-(3,-27)-and-C-cuts-the-x-axis-at-the-point-A-Edexcel-A-Level Maths Pure-Question 2-2011-Paper 1.png

Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x^2(9 - 2x) There is a minimum at the origin, a maximum at the point (3, 27) and C cuts t... show full transcript

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x^2(9 - 2x) There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A - Edexcel - A-Level Maths Pure - Question 2 - 2011 - Paper 1

Step 1

Write down the coordinates of the point A.

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Answer

To find the coordinates of point A where the curve C cuts the x-axis, we set the function equal to zero:

$f(x) = x^2(9 - 2x) = 0$

This implies:

$x^2 = 0 \Rightarrow x = 0$ (which gives the minimum at the origin)
$9 - 2x = 0 \Rightarrow x = 4.5$ (which gives the x-intercept) Thus, the coordinates of point A are (4.5, 0).

Step 2

On separate diagrams sketch the curve with equation y = f(x + 3).

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Answer

This equation represents a horizontal translation of the original curve C by 3 units to the left. The maximum point of the original curve at (3, 27) will shift to (-0.5, 27). The key points to mark on the sketch include:

Minimum point at (-3, 0)
Maximum point at (-0.5, 27)
X-intercepts will be at approximately (-0.5, 0) and (-6.5, 0).

Step 3

On separate diagrams sketch the curve with equation y = f(3x).

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Answer

This equation compresses the x-axis by a factor of 1/3. The maximum point at (3, 27) will move to (1, 27).

The minimum point remains at the origin (0, 0).
The new x-intercepts can be calculated by setting f(3x) = 0, leading to two intercepts. Key points to mark include:
Maximum at (1, 27)
Minimum at (0, 0)

Step 4

Write down the value of k.

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Answer

To find the value of k where the maximum point shifts to (3, 10), we use the original maximum value at (3, 27). Since the maximum has dropped from 27 to 10, we have:

$10 = 27 + k$

Solving for k gives: $k = 10 - 27 = -17$ Thus, the value of k is -17.

Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x^2(9 - 2x) There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A - Edexcel - A-Level Maths Pure - Question 2 - 2011 - Paper 1

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x^2(9 - 2x) There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A - Edexcel - A-Level Maths Pure - Question 2 - 2011 - Paper 1

Write down the coordinates of the point A.

On separate diagrams sketch the curve with equation y = f(x + 3).

On separate diagrams sketch the curve with equation y = f(3x).

Write down the value of k.

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