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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 $y = f'(x)$ [2 marks] [2... show full transcript
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 $y = f(-x)$
Answer
To sketch the graph of , we reflect the graph of across the y-axis. This means that for every point on the original graph, we will plot the point .
The points of interest from the original graph are:
- reflects to
- reflects to (no change)
- reflects to
- reflects to
Thus, the new points to plot are , , , and . Connect these points to complete the reflection.
Step 2
Sketch the graph of $y = 2f(x) - 4$
Answer
For the function , we first apply a vertical stretch by a factor of 2, followed by a downward translation of 4 units.
-
Vertical Stretch: Multiply the y-coordinates of each point from the original graph by 2.
- becomes
- becomes
- stays at
- becomes
-
Downward Translation: Subtract 4 from each y-coordinate:
- becomes
- becomes
- becomes
- becomes
Plot these new points, connecting them according to the transformations applied.
Step 3
Sketch the graph of $y = f'(x)$
Answer
To sketch the graph of the derivative , we need to analyze the slopes between key points of the original function .
- Between points and , the slope is positive, therefore .
- The slope is zero at where the function has a horizontal tangent.
- Between and , the slope is also positive as the function is increasing.
- After , the function appears to flatten out, indicating that the derivative approaches zero.
The graph consists of:
- A positive line segment between and
- A horizontal line at between and
- Another positive line segment between and
- A horizontal line at again after .
Mark these sections clearly, indicating where is positive and where it changes from positive to zero.