Figure 1 shows part of the curve with equation $y = f(x)$, $x \\in \\mathbb{R}$, where $f$ is an increasing function of $x$. The curve passes through the points $P(0, -2)$ and $(3, 0)$ 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.

A-Level Edexcel Maths Pure Applications of Differentiation: Figure 1 shows part of the curve

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

Step 1

Sketch the curve with equation $y = |f(x)|$

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Answer

To sketch the curve for $y = |f(x)|$ , we need to reflect the part of the graph that lies below the x-axis, since absolute values transform negative values into positive counterparts. The original curve dips at $P(0, -2)$ and crosses the x-axis at $(3, 0)$ . Therefore, the reflection will pass through points $(0, 2)$ and $(3, 0)$ , maintaining the x-coordinates while modifying the y-coordinates. This results in a graph that has a sharp point (cusp) at the origin and reflects the original curve appropriately. The coordinates to note here are $(0, 2)$ and $(3, 0)$ .

Step 2

Sketch the curve with equation $y = f(x)$

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Answer

In part (b), we sketch the original function $y = f(x)$ . As given, this function is an increasing function, passing through $(0, -2)$ and $(3, 0)$ . The curve will start below the x-axis and slope upwards towards and beyond $(3, 0)$ . We clearly mark the axes intersections using points $(-2, 0)$ and $(0, 3)$ . Hence, the coordinates to highlight are $(0, -2)$ and $(3, 0)$ .

Step 3

Sketch the curve with equation $y = f(3x)$

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Answer

For $y = f(3x)$ , we compress the graph horizontally by a factor of 3. As a result, the xcrossings will move closer to the origin compared to $y = f(x)$ . Specifically, this transforms the original intercepts: $(0, -2)$ results in an x-intercept at the new coordinate; for the x intercept at $(3,0)$ , we get $(1,0)$ . Meanwhile, the original curve's shape remains the same. The key coordinates to identify in this transformed graph are $(0, -1)$ and $(1, 0)$ .

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

Sketch the curve with equation $y = |f(x)|$

Sketch the curve with equation $y = f(x)$

Sketch the curve with equation $y = f(3x)$

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