A-Level Edexcel Maths Pure Applications of Differentiation: 8. (a) Sketch the graph of $y =

8. (a) Sketch the graph of $y = 7^x$, $x \\in \\mathbb{R}$, showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 10 - 2011 - Paper 3

Question 10

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8. (a) Sketch the graph of $y = 7^x$, $x \\in \\mathbb{R}$, showing the coordinates of any points at which the graph crosses the axes. (b) Solve the equation $$7^x... show full transcript

Worked Solution & Example Answer:8. (a) Sketch the graph of $y = 7^x$, $x \\in \\mathbb{R}$, showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 10 - 2011 - Paper 3

Step 1

Sketch the graph of $y = 7^x$

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Answer

To sketch the graph of the function $y = 7^x$ , we start by identifying key points:

Intercepts:
- The graph intersects the y-axis at the point (0, 1) since $7^0 = 1$ .
- The graph does not cross the x-axis since $y = 7^x > 0$ for all real x.
Behavior:
- As $x o -\\infty$ , $y = 7^x$ approaches 0, and as $x o +\\infty$ , $y$ increases without bound.
Sketch:
- Draw a smooth curve starting from the vicinity of (0, 1), approaching the x-axis from above as it moves left, and rising steeply as it moves right.

Step 2

Solve the equation $7^x - 4(7) + 3 = 0$

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Answer

To solve the equation, start by simplifying it:

$7^x - 28 + 3 = 0$ $7^x - 25 = 0$

This gives: $7^x = 25$ Taking the logarithm on both sides: $x = \log_7(25)$ Using the change of base formula: $x = \frac{\log(25)}{\log(7)}$ Using a calculator to find the logarithm values: $\log(25) \approx 1.39794, \quad \log(7) \approx 0.84510$ Then: $x \approx \frac{1.39794}{0.84510} \approx 1.65$

Therefore, the solution to two decimal places is: $x \approx 1.65$ .

8. (a) Sketch the graph of $y = 7^x$, $x \\in \\mathbb{R}$, showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 10 - 2011 - Paper 3

Worked Solution & Example Answer:8. (a) Sketch the graph of $y = 7^x$, $x \\in \\mathbb{R}$, showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 10 - 2011 - Paper 3

Sketch the graph of $y = 7^x$

Solve the equation $7^x - 4(7) + 3 = 0$

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