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Figure 1 shows the graph of y = f(x), 1 < x < 9 - Edexcel - A-Level Maths Pure - Question 4 - 2009 - Paper 2
Question 4
Figure 1 shows the graph of y = f(x), 1 < x < 9. The points T(3, 5) and S(7, 2) are turning points on the graph. Sketch, on separate diagrams, the graphs of (a) y... show full transcript
Worked Solution & Example Answer:Figure 1 shows the graph of y = f(x), 1 < x < 9 - Edexcel - A-Level Maths Pure - Question 4 - 2009 - Paper 2
Step 1
Sketch the graph of y = 2f(x) - 4
Answer
To sketch the graph of y = 2f(x) - 4, we begin by analyzing the original graph of f(x). The transformation involves a vertical stretch by a factor of 2 followed by a downward shift of 4 units.
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Starting with the turning points of f(x):
- T(3, 5) becomes T'(3, 2) after applying the transformations (5 * 2 - 4 = 6 - 4 = 2).
- S(7, 2) becomes S'(7, 0) after transformations (2 * 2 - 4 = 4 - 4 = 0).
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The new turning points T'(3, 2) and S'(7, 0) should be indicated on the graph.
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Sketch the transformed curve, ensuring it reflects the appropriate shape based on the original function.
Step 2
Sketch the graph of y = |f(x)|
Answer
For the graph y = |f(x)|, we need to reflect any portion of the original graph that is below the x-axis across the x-axis while keeping the portions above the x-axis unchanged.
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From the original graph, T(3, 5) remains unchanged since it is above the x-axis.
- Mark the point T(3, 5).
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The point S(7, 2) also remains unchanged, as it is above the x-axis.
- Mark the point S(7, 2).
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The region between x = 3 and x = 7, where f(x) dips below the x-axis, needs to be reflected above it. This includes modifying the curve appropriately so that no part of the graph is below the x-axis.
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Indicate the turning points (3, 5) and (7, 2) on the graph, representing the appropriate shape of the absolute value function.