Figure 1 shows a sketch of the curve C with equation $y = f(x)$. The curve C passes through the origin and through $(6, 0)$. The curve C has a minimum at the point $(3, -1)$. On separate diagrams, sketch the curve with equation (a) $y = f(2x)$, (b) $y = -f(x)$, (c) $y = f(x + p)$, where $p$ is a constant and $0 < p < 3$. On each diagram show the coordinates of any points where the curve intersects the x-axis and of any minimum or maximum points.

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Figure 1 shows a sketch of the curve C with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2011 - Paper 1

Question 10

Figure 1 shows a sketch of the curve C with equation $y = f(x)$. The curve C passes through the origin and through $(6, 0)$. The curve C has a minimum at the point $... 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 10 - 2011 - Paper 1

Step 1

a) $y = f(2x)$

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Answer

For the transformation $y = f(2x)$ , the graph is compressed horizontally by a factor of 2. The shape remains U-shaped and will still pass through the point (0,0) due to the original function's behavior. Thus, the coordinates where the graph intersects the x-axis are:

(0, 0)
$igg( rac{3}{2}, 0 igg)$
(1.5, -1)

Final output:

Shape: U-shaped through the origin.
Coordinates at x-intercepts: (0,0), (1.5,-1).

Step 2

b) $y = -f(x)$

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Answer

This transformation reflects the graph of $y = f(x)$ over the x-axis. The curve will peak at (3, 1) while passing through the points (0, 0) and (6, 0) in a downward direction. Therefore, the coordinates where the graph intersects the x-axis remain unchanged as:

(0, 0)
(6, 0)
(3, 1)

Final output:

Shape: U-shaped downwards.
Coordinates at x-intercepts: (0,0), (6,0).

Step 3

c) $y = f(x + p)$, where $0 < p < 3$

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Answer

This transformation translates the graph to the left by $p$ units, where $p$ is a positive constant less than 3. The minimum point will shift from (3, -1) to $(3-p, -1)$ . The curve will still retain its U-shape but will be positioned differently in terms of x-intercepts. We determine the x-intercepts:

$igg( 3 - p, 0 igg)$
(6 - p, 0)
(3 - p, -1)

Final output:

Shape: U-shaped, not through origin.
Coordinates of minimum at: $(3 - p, -1)$ ; X-intercepts at: $(3-p,0)$ and $(6-p,0)$ .

Figure 1 shows a sketch of the curve C with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2011 - Paper 1

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve C with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2011 - Paper 1

a) $y = f(2x)$

b) $y = -f(x)$

c) $y = f(x + p)$, where $0 < p < 3$

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