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

Question 4

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Figure 1 shows part of the curve with equation $y = f(x)$, $x ext{ } orall ext{ } ext{ } ext{ }$, where $f$ is an increasing function of $x$. The curve passes t... show full transcript

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

Step 1

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

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Answer

To sketch the graph of $y = |f(x)|$ , we begin by noting that the function $f(x)$ is defined such that it is increasing and intersects the x-axis at the point $Q(3,0)$ , where $f(3) = 0$ . The point $P(0, -2)$ indicates that $f(0) = -2$ .

Since we are taking the absolute value, the portion of the graph where $f(x) < 0$ will reflect above the x-axis. This means:

The section between $x=0$ and $x=3$ will reflect above the x-axis.
The curve will continue increasing past $(3,0)$ , maintaining the shape and characteristics from point (3,0) onward.

Thus, we identify key points:

The curve will meet the y-axis at $(0, 2)$ (since it's the reflection of the point $(0, -2)$ ).
It passes through $(3, 0)$ , indicating no reflection there.

Therefore, the curve meets the axes at the points $(0, 2)$ and $(3, 0)$ .

Step 2

b) $y = f(x)$

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Answer

To sketch the graph of $y = f(x)$ , we directly use the properties given in the question. We note that:

The curve is increasing.
It passes through the points $P(0, -2)$ and $Q(3, 0)$ .

The sketch will show:

A continuous curve starting from $(0, -2)$ , increasing to $(3, 0)$ , and continuing to rise beyond.
The graph will have a cusp at points where it transitions from negative to positive (around the x-axis).
It is essential to reflect the correct curvature, indicating the increasing nature at all points.

The intersection points with the axes are $(0, -2)$ and $(3, 0)$ .

Step 3

c) $y = f(3x)$

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Answer

To sketch the graph of $y = f(3x)$ , we analyze the effect of the transformation on the original function:

The graph will compress horizontally by a factor of 3.
Key points will transform:
- The point $(0, -2)$ transforms to $(0, -2)$ as $f(3 imes 0) = f(0) = -2$ .
- The point $(3, 0)$ becomes $(1, 0)$ as $f(3 imes 1) = f(3) = 0$ .

Hence, the sketch will show the endpoints at:

$(0, -2)$ and $(1, 0)$ , maintaining the increasing nature and correct curvature of the graph.

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

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

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

b) $y = f(x)$

c) $y = f(3x)$

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