The graph of $y = f(x)$ is shown below - AQA - A-Level Maths: Pure - Question 6 - 2020 - Paper 3
Question 6
The graph of $y = f(x)$ is shown below.
Sketch the graph of $y = f(-x)$
Sketch the graph of $y = 2f(x) - 4$
Sketch the graph of $y = f'(x)$
Worked Solution & Example Answer:The graph of $y = f(x)$ is shown below - AQA - A-Level Maths: Pure - Question 6 - 2020 - Paper 3
Step 1
Sketch the graph of $y = f(-x)$
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Answer
To sketch the graph of y=f(−x), you will reflect the given graph across the y-axis.
Identify key points: The original graph contains the points:
(−1,0)
(0,2)
(2,6)
(−2,−2)
Reflect points across the y-axis:
(−1,0)→(1,0)
(0,2)→(0,2)
(2,6)→(−2,6)
(−2,−2)→(2,−2)
Draw the reflected graph: Connect the reflected points smoothly. Ensure the overall shape reflects the original graph while maintaining the y-coordinates.
Step 2
Sketch the graph of $y = 2f(x) - 4$
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Answer
To sketch the graph of y=2f(x)−4, follow these steps:
Vertical stretch: Multiply the y-values of the original graph by 2. For example:
(−1,0)→(−1,0)
(0,2)→(0,4)
(2,6)→(2,12)
(−2,−2)→(−2,−4)
Vertical shift down: Subtract 4 from all y-values. Updating those points:
(−1,0)→(−1,−4)
(0,4)→(0,0)
(2,12)→(2,8)
(−2,−4)→(−2,−8)
Draw the new graph: Connect these points smoothly to show the transformed graph.
Step 3
Sketch the graph of $y = f'(x)$
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Answer
To sketch y=f′(x) which represents the derivative of f(x), consider the following steps:
Identify intervals of increase and decrease:
The function f(x) increases from (−1,0) and (0,2), hence f′(x)>0 in these intervals.
The function decreases in the interval (2,+extinfinity), hence f′(x)<0 here.
Identify critical points: From the graph, find the points where f(x) has horizontal tangents and changes direction, which indicates where f′(x)=0. For instance, at (0,2) and where the curve begins to decrease.
Sketch the derivative:
For x<−2, f′(x)=0 leading to a horizontal line.
For −2<x<0, f′(x) is positive, represented by above the x-axis.
