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2. (a) Sketch the curve with equation y = 4^ x stating any points of intersection with the coordinate axes - Edexcel - A-Level Maths: Pure - Question 4 - 2022 - Paper 2
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2. (a) Sketch the curve with equation y = 4^ x stating any points of intersection with the coordinate axes. (b) Solve 4^x = 100 giving your answer to 2 decima... show full transcript
Worked Solution & Example Answer:2. (a) Sketch the curve with equation y = 4^ x stating any points of intersection with the coordinate axes - Edexcel - A-Level Maths: Pure - Question 4 - 2022 - Paper 2
Step 1
Sketch the curve with equation y = 4^x stating any points of intersection with the coordinate axes.
Answer
To sketch the curve defined by the equation , we observe that:
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Behavior of the Curve:
- The curve is an exponential function which rises steeply as increases. It is defined for all real and passes through the point (0, 1) since .
- As approaches negative infinity, approaches 0, meaning the curve never quite touches the x-axis (horizontal asymptote).
- The curve lies entirely in quadrants I and II since is always positive.
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Points of Intersection:
- X-axis Intersection: There is no intersection with the x-axis as has no solution in this context.
- Y-axis Intersection: The curve intersects the y-axis at the point (0, 1).
Step 2
Solve 4^x = 100 giving your answer to 2 decimal places.
Answer
To solve the equation , we can proceed as follows:
- Taking Logarithms:
- Apply logarithm on both sides to make the exponent manageable. Using natural logarithm or logarithm base 10, the equation can be transformed: