A-Level Edexcel Maths Pure Trigonometric Equations: Sketch the graph of $y = 7^x$,

Sketch the graph of $y = 7^x$, $x eq ext{R}$, showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 2 - 2010 - Paper 4

Question 2

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Sketch the graph of $y = 7^x$, $x eq ext{R}$, showing the coordinates of any points at which the graph crosses the axes. Solve the equation $7^{2x} - 4(7^x) + ... show full transcript

Worked Solution & Example Answer:Sketch the graph of $y = 7^x$, $x eq ext{R}$, showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 2 - 2010 - Paper 4

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 need to analyze its behavior:

Identify Axes Intercepts:
- The graph crosses the y-axis when $x = 0$ : $y = 7^0 = 1$ Therefore, the point is $(0, 1)$ .
- The graph does not cross the x-axis, as the function never reaches zero for any real $x$ .
General Behavior:
- As $x$ approaches negative infinity, $y$ approaches 0 but never touches it.
- As $x$ increases, $y$ increases rapidly because it is an exponential function.
Sketch the Graph:
- The graph starts near the x-axis and passes through the point $(0, 1)$ , rising steeply as $x$ increases.

Step 2

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

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Answer

To solve the equation, we can use substitution:

Substitution: Let $u = 7^x$ . Then, the equation becomes: $u^2 - 4u + 3 = 0$
Factor the Quadratic: This can be factored as: $(u - 3)(u - 1) = 0$
Solve for $u$ :
- Setting each factor to zero gives:
  - $u - 3 = 0 ightarrow u = 3$
  - $u - 1 = 0 ightarrow u = 1$
Back Substitute for $x$ :
- For $u = 3$ :

ightarrow x = rac{ ext{log}(3)}{ ext{log}(7)} \ ext{Calculate: } x \ ≈ 0.564.$$

For $u = 1$ :

ightarrow x = 0 \ ext{Exact value: } x = 0.$$

Final Answers:
- Therefore, the solutions to the original equation are: $x ext{ (to 2 decimal places)}: \\ x = 0.56 ext{ and } x = 0.$

Sketch the graph of $y = 7^x$, $x eq ext{R}$, showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 2 - 2010 - Paper 4

Worked Solution & Example Answer:Sketch the graph of $y = 7^x$, $x eq ext{R}$, showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 2 - 2010 - Paper 4

Sketch the graph of $y = 7^x$

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

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