Figure 1 shows part of the curve with equation $y = f(x)$, $x ext{ in } eal$, where $f$ is an increasing function of $x$. The curve passes through the points $P(0, -2)$ and $Q(3, 0)$, as shown. In separate diagrams, sketch the curve with equation (a) $y = |f(x)|$. (b) $y = f(x)$. (c) $y = f(3x)$. Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

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Figure 1 shows part of the curve with equation $y = f(x)$, $x ext{ in } eal$, where $f$ is an increasing function of $x$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

Question 5

Figure 1 shows part of the curve with equation $y = f(x)$, $x ext{ in } eal$, where $f$ is an increasing function of $x$. The curve passes through the points $P(0,... show full transcript

Worked Solution & Example Answer:Figure 1 shows part of the curve with equation $y = f(x)$, $x ext{ in } eal$, where $f$ is an increasing function of $x$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

Step 1

(a) $y = |f(x)|$

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Answer

To sketch the graph of $y = |f(x)|$ , we need to reflect any parts of the graph of $f(x)$ that lie below the x-axis upwards.

Start from point $P(0, -2)$ . Since it is below the x-axis, reflect it to $(0, 2)$ .
The point $Q(3, 0)$ remains unchanged as it is on the x-axis.
Draw the curve, ensuring that the transition around the x-axis is smooth, resulting in a cusp at $(0, 2)$ .
Clearly mark the points $(0, 2)$ and $(3, 0)$ on the axes.

Step 2

(b) $y = f(x)$

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Answer

For the graph of $y = f(x)$ :

Begin with the original form of the function, recognizing that the points given are $P(0, -2)$ and $Q(3, 0)$ .
Plot these points.
The curve must be drawn smoothly, maintaining an increasing nature as stipulated in the question, reflecting accurately the increasing function behavior around those points.
The coordinates at which the curve meets the axes are $(-2, 0)$ and $(0, 3)$ , making sure these are fully represented.

Step 3

(c) $y = f(3x)$

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Answer

To sketch $y = f(3x)$ , we apply a horizontal compression by a factor of 3:

Original points scale appropriately. For $x = 0$ , $y$ remains $f(0)$ , which is $-2$ .
For all x-values, apply $x/3$ to compute the points accordingly. So $f(0)$ gives $(0, -2)$ .
For the x-intercept $Q(3, 0)$ , transform to $(1, 0)$ under this new equation.
Mark the new point of intersection at the x-axis $(1, 0)$ and the curve should be drawn reflecting the same general shape, keeping its characteristics.

Figure 1 shows part of the curve with equation $y = f(x)$, $x ext{ in } eal$, where $f$ is an increasing function of $x$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

Worked Solution & Example Answer:Figure 1 shows part of the curve with equation $y = f(x)$, $x ext{ in } eal$, where $f$ is an increasing function of $x$ - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 4

(a) $y = |f(x)|$

(b) $y = f(x)$

(c) $y = f(3x)$

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