Given the function: $$f(x) = 2x^2 + 4x + 9$$ where $x \in \mathbb{R}$. (a) Write $f(x)$ in the form $\alpha(x + b)^2 + c$, where $a$, $b$, and $c$ are integers to be found. (b) Sketch the curve with equation $y = f(x)$ showing any points of intersection with the coordinate axes and the coordinates of any turning point. (c) (i) Describe fully the transformation that maps the curve with equation $y = f(x)$ onto the curve with equation $y = g(x)$ where $g(x) = 2(x - 2)^2 - 4x - 3$, $x \in \mathbb{R}$ (ii) Find the range of the function $$h(x) = \frac{21}{2x^2 + 4x + 9}, \quad x \in \mathbb{R}$$

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Given the function: $$f(x) = 2x^2 + 4x + 9$$ where $x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 5 - 2019 - Paper 1

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Given the function: $$f(x) = 2x^2 + 4x + 9$$ where $x \in \mathbb{R}$. (a) Write $f(x)$ in the form $\alpha(x + b)^2 + c$, where $a$, $b$, and $c$ are integers... show full transcript

Worked Solution & Example Answer:Given the function: $$f(x) = 2x^2 + 4x + 9$$ where $x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 5 - 2019 - Paper 1

Step 1

Write $f(x)$ in the form $\alpha(x + b)^2 + c$

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Answer

To convert $f(x)$ into the required form, we complete the square:

Start with the original function:
$f(x) = 2x^2 + 4x + 9$
Factor out the leading coefficient from the quadratic and linear terms:
$= 2(x^2 + 2x) + 9$
Complete the square inside the parentheses:
$= 2[(x + 1)^2 - 1] + 9$
Simplify:
$= 2(x + 1)^2 - 2 + 9$
$= 2(x + 1)^2 + 7$
Thus, we have $a = 2$ , $b = 1$ , and $c = 7$ .

Step 2

Sketch the curve with equation $y = f(x)$

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Answer

Identify the vertex and the minimum:
The vertex from the completed square is at $(-1, 7)$ .
The function is U-shaped, indicating it opens upwards.
Find x-intercepts:
Set $f(x) = 0$ :
$2x^2 + 4x + 9 = 0$
This has no real solutions since the discriminant is negative.
Find y-intercept:
Set $x = 0$ :
$f(0) = 9$ , hence the y-intercept is $(0, 9)$ .
Draw the curve:
Plot the vertex and y-intercept. There are no x-intercepts.

Step 3

Describe fully the transformation that maps $y = f(x)$ onto $y = g(x)$

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Answer

Base function: Start with $y = f(x)$ .
Horizontal transformation: Reflect the graph over the line $x = 2$ (shift right by 2) resulting in $g(x) = f(x - 2)$ .
Vertical transformation: Stretch the graph vertically by a factor of 2 and shift down by 3. Therefore, $g(x) = 2f(x - 2) - 3$ .
This describes the complete mapping of $y = f(x)$ to $y = g(x)$ .

Step 4

Find the range of the function $h(x)$

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Answer

The function can be rewritten as: $h(x) = \frac{21}{2(x^2 + 2x + 9/2)}$ .
The minimum value of the denominator occurs at the vertex of the quadratic:
Completing the square gives:
$= 2((x + 1)^2 + 8)$
The minimum value of $h(x)$ therefore happens when $(x + 1)^2 = 0$ :
$\Rightarrow \text{Minimum value is: } \frac{21}{2(8)} = \frac{21}{16}$ .
The maximum value of $h(x)$ is 3 as $x \to \pm \infty$ .
Thus the range of $h(x)$ is: $0 < h(x) < 3$ .

Given the function: $$f(x) = 2x^2 + 4x + 9$$ where $x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 5 - 2019 - Paper 1

Worked Solution & Example Answer:Given the function: $$f(x) = 2x^2 + 4x + 9$$ where $x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 5 - 2019 - Paper 1

Write $f(x)$ in the form $\alpha(x + b)^2 + c$

Sketch the curve with equation $y = f(x)$

Describe fully the transformation that maps $y = f(x)$ onto $y = g(x)$

Find the range of the function $h(x)$

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