Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x³(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 Equation of a Straight Line: 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) = x³(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 7 - 2012 - Paper 2

Question 7

Figure-1-shows-a-sketch-of-the-curve-C-with-equation-y-=-f(x)-where-f(x)-=-x³(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 7-2012-Paper 2.png

Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x³(9 – 2x) There is a minimum at the origin, a maximum at the point (3, 27) and C cuts th... show full transcript

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x³(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 7 - 2012 - Paper 2

Step 1

Write down the coordinates of the point A.

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Answer

The coordinates of point A, where the curve C crosses the x-axis, can be found by setting the function f(x) to zero.

Setting the equation: $f(x) = x^3(9 - 2x) = 0$ This gives us roots at x = 0 and x = 4. Therefore, the coordinates of point A are (4, 0).

Step 2

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

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Answer

In this transformation, shifting the graph left by 3 units move the maximum point from (3, 27) to (0, 27). The new coordinates would be (0, 27) and the curve maintains its shape and orientation, with intersections with the x-axis at (–3, 0) and points clearly marked.

Sketch:

Maximum at (0, 27)
X-axis crossing at (–3, 0)

Step 3

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

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Answer

In this case, the transformation compresses the graph horizontally by a factor of 3. The maximum point moves from (3, 27) to (1, 27), and the overall dimensions of the graph change. Points where the curve intersects the x-axis will also shift accordingly.

Sketch:

Maximum at (1, 27)
X-axis crossings indicate new positions for any previously defined points, such as (1, 0).

Step 4

Write down the value of k.

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Answer

Given that the maximum point of the curve y = f(x) + k is now at (3, 10), and originally the maximum at (3, 27), we find k by solving:

$27 + k = 10$ This results in: $k = 10 - 27$ Thus, we can conclude that: $k = -17$

Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x³(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 7 - 2012 - Paper 2

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve C with equation y = f(x) where f(x) = x³(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 7 - 2012 - Paper 2

Write down the coordinates of the point A.

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

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

Write down the value of k.

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