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8 (a) Sketch the graph of $y = \frac{1}{x^2}$ - AQA - A-Level Maths Pure - Question 8 - 2022 - Paper 2
Question 8
8 (a) Sketch the graph of $y = \frac{1}{x^2}$. 8 (b) The graph of $y = \frac{1}{x^2}$ can be transformed onto the graph of $y = \frac{9}{x^2}$ using a stretch in on... show full transcript
Worked Solution & Example Answer:8 (a) Sketch the graph of $y = \frac{1}{x^2}$ - AQA - A-Level Maths Pure - Question 8 - 2022 - Paper 2
Step 1
Sketch the graph of $y = \frac{1}{x^2}$
Answer
To sketch the graph of the function , we note that it is a rational function that has vertical asymptotes at and approaches zero as approaches both infinity and negative infinity. The graph exists only in the first and second quadrants, where is positive. The shape is a decreasing curve towards the asymptotes, which means it never touches the axes.
Step 2
The graph of $y = \frac{1}{x^2}$ can be transformed onto the graph of $y = \frac{9}{x^2}$
Answer
To determine the nature of the transformation from to , we recognize that this represents a vertical stretch.
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Beth's Argument: A stretch in the -direction by a scale factor of means that for every value on the original graph, we multiply it by . Thus, the new graph would have higher values at all corresponding values.
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Paul's Argument: A stretch in the -direction would imply modifying the values instead, which is not the case here.
Therefore, Beth is correct as the transformation indeed stretches the graph vertically by a scale factor of , while Paul is incorrect.