A-Level Edexcel Maths Pure Quadratics: Sketch the graph of $y = 3^x$,

Sketch the graph of $y = 3^x$, $x \in \mathbb{R}$ showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 1

Question 10

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Sketch the graph of $y = 3^x$, $x \in \mathbb{R}$ showing the coordinates of any points at which the graph crosses the axes. Use algebra to solve the equation $... show full transcript

Worked Solution & Example Answer:Sketch the graph of $y = 3^x$, $x \in \mathbb{R}$ showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 1

Step 1

Sketch the graph of $y = 3^x$

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Answer

To sketch the graph of the function $y = 3^x$ , we start by identifying key coordinates.

Intercepts:
- Y-Intercept: When $x = 0$ , $y = 3^0 = 1$ . Thus, the graph crosses the y-axis at the point $(0, 1)$ .
- X-Intercept: The function does not cross the x-axis because $3^x > 0$ for all real $x$ .
Behavior:
- For large positive $x$ , the function $y = 3^x$ will increase rapidly.
- For large negative $x$ , the function approaches $0$ but never touches the x-axis.

The graph is an increasing exponential curve starting from a value greater than zero as $x$ approaches negative infinity and rising steeply as $x$ becomes positive.

Step 2

Use algebra to solve the equation $3^{2x} - 9(3^x) + 18 = 0$

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Answer

To solve the equation, we can make a substitution:

Let $y = 3^x$ , then $3^{2x} = y^2$ . The equation becomes:

$y^2 - 9y + 18 = 0$

This is a quadratic equation in standard form. We can factor it:

$(y - 6)(y - 3) = 0$

Setting each factor to zero gives:

$y - 6 = 0 \implies y = 6$
$y - 3 = 0 \implies y = 3$

Now, we can revert back to $x$ :

From $y = 6$ :
$3^x = 6 \implies x = \log_3{6}$
Approximating gives $x \approx 1.631$ .
From $y = 3$ :
$3^x = 3 \implies x = 1$ .

Thus, the solutions are approximately:

$x \approx 1.63$
$x = 1$ .

Sketch the graph of $y = 3^x$, $x \in \mathbb{R}$ showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 1

Worked Solution & Example Answer:Sketch the graph of $y = 3^x$, $x \in \mathbb{R}$ showing the coordinates of any points at which the graph crosses the axes - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 1

Sketch the graph of $y = 3^x$

Use algebra to solve the equation $3^{2x} - 9(3^x) + 18 = 0$

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