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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 :
- The function has asymptotes at and , which means it will approach the axes but never touch them.
- The graph is symmetrical and will lie in the first and second quadrants because it only takes positive values for except for when equals zero.
- The shape of the curve approaches infinity as approaches 0 from either side, and as increases, approaches 0.
Step 2
The graph of $y = \frac{1}{x^2}$ can be transformed onto the graph of $y = \frac{9}{x^2}$ using a stretch in one direction.
Answer
To determine the correct direction of the stretch:
- The equation can be seen as a stretch of the graph of in the -direction by a scale factor of 9. This is because multiplying the output by 9 stretches the curve away from the x-axis.
- Conversely, it could also be viewed as a stretch in the -direction by a scale factor of since can be rewritten as .
- Therefore, both Beth and Paul are correct, as the graph can be expressed in both stretch forms.