Given that $f(x) = x^2 - ax - 18, \, x \geq 0,$ a) express $f(x)$ in the form $(x - \alpha)^2 + b$, where $\alpha$ and $b$ are integers. The curve $C$ with equation $y = f(x), \, x \geq 0,$ meets the $y$-axis at $P$ and has a minimum point at $Q.$ b) Sketch the graph of $C$, showing the coordinates of $P$ and $Q.$ The line $y = 41$ meets $C$ at the point $R.$ c) Find the $x$-coordinate of $R$, giving your answer in the form $p + q\sqrt{2}$, where $p$ and $q$ are integers.

A-Level Edexcel Maths Pure Basic Trigonometry: Given that $f(x) = x^2 - ax

Given that $f(x) = x^2 - ax - 18, \, x \geq 0,$ a) express $f(x)$ in the form $(x - \alpha)^2 + b$, where $\alpha$ and $b$ are integers - Edexcel - A-Level Maths Pure - Question 2 - 2016 - Paper 2

Question 2

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Given that $f(x) = x^2 - ax - 18, \, x \geq 0,$ a) express $f(x)$ in the form $(x - \alpha)^2 + b$, where $\alpha$ and $b$ are integers. The curve $C$ with equati... show full transcript

Worked Solution & Example Answer:Given that $f(x) = x^2 - ax - 18, \, x \geq 0,$ a) express $f(x)$ in the form $(x - \alpha)^2 + b$, where $\alpha$ and $b$ are integers - Edexcel - A-Level Maths Pure - Question 2 - 2016 - Paper 2

Step 1

a) express $f(x)$ in the form $(x - \alpha)^2 + b$

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Answer

To express $f(x) = x^2 - ax - 18$ in the form $(x - \alpha)^2 + b$ , we need to complete the square.

Start by rewriting the quadratic:
$f(x) = x^2 - ax - 18$
Take half of the coefficient of $x$ , square it and add/subtract that value:
$f(x) = (x - \frac{a}{2})^2 - \frac{a^2}{4} - 18$
Combine the constants to achieve the desired form, where
$b = -\frac{a^2}{4} - 18$
This gives us the expression in the required form.

Step 2

b) Sketch the graph of $C$, showing the coordinates of $P$ and $Q$

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Answer

To sketch the graph of $C$ , consider the following:

The vertex of the parabola is where the minimum point $Q$ $Q$ is located.
- The coordinates of $Q$ can be obtained from $x = \frac{a}{2}$ .
- Calculate $y$ at this $x$ to find $Q$ .
The point $P$ $P$ is found where the curve meets the $y$ $y$ -axis, that is when $x = 0$ $x = 0$ :
- For $x = 0$ , $f(0) = -18$ , giving $P(0, -18)$ .
Sketch the parabola ensuring it opens upwards and passes through points $P$ and $Q$ , labeling these points accordingly.

Step 3

c) Find the $x$-coordinate of $R$, giving your answer in the form $p + q\sqrt{2}$

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Answer

To find the $x$ -coordinate of point $R$ where the line $y = 41$ intersects the curve:

Set the equation of the curve equal to 41:
$x^2 - ax - 18 = 41$
This simplifies to
$x^2 - ax - 59 = 0$ .
Use the quadratic formula to solve for $x$ :
$x = \frac{a \pm \sqrt{a^2 + 236}}{2}$
The term under the square root can be simplified to find the exact expression, ensuring the result can be expressed as $p + q\sqrt{2}$ $p + q 2$ .
- Specifically, since $\sqrt{236} = \sqrt{4 \cdot 59} = 2\sqrt{59}$ , thus completing the solution.

Given that $f(x) = x^2 - ax - 18, \, x \geq 0,$ a) express $f(x)$ in the form $(x - \alpha)^2 + b$, where $\alpha$ and $b$ are integers - Edexcel - A-Level Maths Pure - Question 2 - 2016 - Paper 2

Worked Solution & Example Answer:Given that $f(x) = x^2 - ax - 18, \, x \geq 0,$ a) express $f(x)$ in the form $(x - \alpha)^2 + b$, where $\alpha$ and $b$ are integers - Edexcel - A-Level Maths Pure - Question 2 - 2016 - Paper 2

a) express $f(x)$ in the form $(x - \alpha)^2 + b$

b) Sketch the graph of $C$, showing the coordinates of $P$ and $Q$

c) Find the $x$-coordinate of $R$, giving your answer in the form $p + q\sqrt{2}$

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