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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 5 - 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... 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 5 - 2022 - Paper 2
Step 1
Sketch the curve with equation $y = 4^x$
Answer
To sketch the curve defined by the equation ( y = 4^x ), follow these steps:
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Identify Key Features: This is an exponential function where the base, 4, is greater than 1. This indicates that the curve will rise steeply as ( x ) increases and approach 0 as ( x ) decreases.
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Intersection with the Axes:
- Y-axis: The curve intersects the y-axis when ( x = 0 ). Plugging in the value gives: [ y = 4^0 = 1 ] Thus, the point of intersection is (0, 1).
- X-axis: The curve does not intersect the x-axis since it approaches 0 as ( y ) but never actually reaches it, indicating that there is no point where ( y = 0 ).
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Drawing the Curve: Begin from the point (0, 1), rise steeply in the first quadrant, and level off in the second quadrant without touching the x-axis. It should be noted that it does not dip below the x-axis.
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Final Touch: Ensure that the curve is smooth, reflecting the continuous nature of exponential functions.
Step 2
Solve $4^x = 100$
Answer
To solve the equation ( 4^x = 100 ), follow these steps:
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Use Logarithms: Taking logarithms on both sides, we have: [ \log(4^x) = \log(100) ]
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Apply Logarithm Properties: Using the property ( \log(a^b) = b \log(a) ), we can rewrite the logarithm as: [ x \log(4) = \log(100) ]
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Isolate x: Solving for ( x ) gives us: [ x = \frac{\log(100)}{\log(4)} ]
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Calculate the Values: Knowing that ( \log(100) = 2 ) (since 100 is ( 10^2 )) and using a scientific calculator for ( \log(4) \approx 0.6021 ), we find: [ x \approx \frac{2}{0.6021} \approx 3.32 ] Thus, the solution rounded to two decimal places is ( x \approx 3.32 ).