Figure 1 shows a sketch of the curve C with equation y = f(x). The curve C passes through the point (−1, 0) and touches the x-axis at the point (2, 0). The curve C has a maximum at the point (0, 4). (a) The equation of the curve C can be written in the form y = x³ + ax² + bx + c where a, b and c are integers. Calculate the values of a, b and c. (b) Sketch the curve with equation y = f^{-1}(x) in the space provided on page 24 Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.

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

Figure 1 shows a sketch of the curve C with equation y = f(x) - Edexcel - A-Level Maths Pure - Question 11 - 2013 - Paper 2

Question 11

Figure 1 shows a sketch of the curve C with equation y = f(x). The curve C passes through the point (−1, 0) and touches the x-axis at the point (2, 0). The curve C... show full transcript

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve C with equation y = f(x) - Edexcel - A-Level Maths Pure - Question 11 - 2013 - Paper 2

Step 1

Calculate the values of a, b and c.

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Answer

To find the coefficients a, b, and c, we start with the polynomial form:

y = x³ + ax² + bx + c.

Using given points:

The curve passes through (−1, 0): $0 = (-1)^3 + a(-1)^2 + b(-1) + c$
Simplifying, we get:
$\ (1)$
The curve touches the x-axis at (2, 0): $0 = (2)^3 + a(2)^2 + b(2) + c$
Which simplifies to:
$\ (2)$
The curve has a maximum at (0, 4): $4 = (0)^3 + a(0)^2 + b(0) + c$
Therefore:
$\ (3)$

Substituting (3) into (1) and (2):

From (1): $a - b = -3 \ (4)$$$
From (2): $4a + 2b = -12 \ (5)$$$

Solving equations (4) and (5):
From (4):
$b = a + 3$
Substituting into (5):
$4a + 2(a + 3) = -12$
Which simplifies to:

\ 6a = -18 \ a = -3$$ From (4): $$-3 - b = -3 \ b = 0$$ Thus, the values are: - a = -3, - b = 0, - c = 4.

Step 2

Sketch the curve with equation y = f^{-1}(x)

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Answer

To sketch the inverse curve y = f^{-1}(x), we first recognize that if the original curve has points (x, y), then the inverse will have points (y, x).

From part (a), we know the original function crosses the axes at:

Point (0, 4)
Point (−1, 0)
Point (2, 0)

Coordinates for the inverse:

Point (4, 0)
Point (0, −1)
Point (0, 2)

Features of the sketch:

Ensure the curve is symmetric about the line y = x.
Identify where the function meets the coordinate axes at indicated points.
Clearly label the intersections and ensure accuracy in the shape reflecting the original function.

Figure 1 shows a sketch of the curve C with equation y = f(x) - Edexcel - A-Level Maths Pure - Question 11 - 2013 - Paper 2

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve C with equation y = f(x) - Edexcel - A-Level Maths Pure - Question 11 - 2013 - Paper 2

Calculate the values of a, b and c.

Sketch the curve with equation y = f^{-1}(x)

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