A-Level Edexcel Maths Pure Equation of a Straight Line: 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 9 - 2011 - Paper 3

Question 9

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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^{... 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 9 - 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 first identify the points where the graph crosses the axes.

The graph crosses the y-axis at the point $(0, 7^0) = (0, 1)$ .
The graph crosses the x-axis when $y = 0$ , but since an exponential function never reaches zero, it does not cross the x-axis.

The general shape of the graph is that it approaches the x-axis but never touches it, rising steeply as $x$ increases. A rough sketch shows an upward curve starting from the point $(0, 1)$ .

Step 2

Solve the equation $7^{2x} - 4(7^x) + 3 = 0$

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Answer

To solve the equation, we can make a substitution. Let $u = 7^x$ , then the equation becomes:

$u^2 - 4u + 3 = 0$

This quadratic can be factored as:

$(u - 3)(u - 1) = 0$

Setting each factor to zero gives:

$u - 3 = 0 \Rightarrow u = 3 \Rightarrow 7^x = 3 \Rightarrow x = \log_7(3)$
$u - 1 = 0 \Rightarrow u = 1 \Rightarrow 7^x = 1 \Rightarrow x = \log_7(1) = 0$

Using the change of base formula, we find:

For $x = \log_7(3)$ , we can express this in decimal form: $x \approx 0.5646$ : rounded to two decimal places is $0.56$ .

Thus, the solutions are:

$x \approx 0.56$
$x = 0$

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 9 - 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 9 - 2011 - Paper 3

Sketch the graph of $y = 7^x$

Solve the equation $7^{2x} - 4(7^x) + 3 = 0$

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