Figure 1 shows part of the curve with equation $y = f(x)$, $x \\in \\mathbb{R}$. The curve passes through the points $Q(0,2)$ and $P(-3,0)$ as shown. (a) Find the value of $f(−3)$. On separate diagrams, sketch the curve with equation (b) $y = f^{+}(x)$, (c) $y = f(|x|) - 2$, (d) $y = 2figg(rac{1}{2}xigg)$. Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

A-Level Edexcel Maths Pure Modelling with Trigonometric Functions: Figure 1 shows part of the curve

Figure 1 shows part of the curve with equation $y = f(x)$, $x \\in \\mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 24 - 2013 - Paper 1

Question 24

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Figure 1 shows part of the curve with equation $y = f(x)$, $x \\in \\mathbb{R}$. The curve passes through the points $Q(0,2)$ and $P(-3,0)$ as shown. (a) Find t... show full transcript

Worked Solution & Example Answer:Figure 1 shows part of the curve with equation $y = f(x)$, $x \\in \\mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 24 - 2013 - Paper 1

Step 1

Find the value of $f(-3)$

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Answer

To find $f(-3)$ , we reference the points on the curve. Since the point $P(-3, 0)$ lies on the curve, it indicates that when $x = -3$ , the value of $y = f(-3) = 0$ . Thus, we conclude:

$f(-3) = 0$

Step 2

Sketch the curve with equation $y = f^{+}(x)$

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Answer

The function $f^{+}(x)$ represents the positive part of $f(x)$ . The sketch of this function presents a graph where any negative values of $f(x)$ are replaced by zero. The important points to mark are $(0,2)$ and $(2,0)$ . The curve should be drawn in such a way that it remains in the positive region above the x-axis, reflecting any negative parts to the x-axis.

Step 3

Sketch the curve with equation $y = f(|x|) - 2$

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Answer

Here, $f(|x|)$ reflects the function about the y-axis due to the absolute value. By subtracting 2, the graph is shifted down by 2 units. Key points to note are $(0,0)$ for $f(0) - 2$ and $(2,0)$ reflects to $(-2,0)$ as well. The curves should mirror each other across the y-axis.

Step 4

Sketch the curve with equation $y = 2f\left(\frac{1}{2}x\right)$

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Answer

This transformation stretches the graph vertically by a factor of 2 and horizontally by a factor of 2. Thus, points that were originally $(x,y)$ will transform to $(2x, 2y)$ . The coordinates to highlight are at $(-6, 0)$ and $(0, 0)$ . The result will show the curve with a wider base and stretched upwards.

Figure 1 shows part of the curve with equation $y = f(x)$, $x \\in \\mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 24 - 2013 - Paper 1

Worked Solution & Example Answer:Figure 1 shows part of the curve with equation $y = f(x)$, $x \\in \\mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 24 - 2013 - Paper 1

Find the value of $f(-3)$

Sketch the curve with equation $y = f^{+}(x)$

Sketch the curve with equation $y = f(|x|) - 2$

Sketch the curve with equation $y = 2f\left(\frac{1}{2}x\right)$

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