Standard +0.8 This question requires understanding of mean value of functions as integrals, combined with systematic tracking of transformations (reflection and translation) and how they affect both the function values and the interval. While each individual concept is standard, synthesizing them correctly in 2 marks demands clear conceptual understanding rather than routine calculation, placing it moderately above average difficulty.
The mean value of the function \(\mathbf{f}\) over the interval \(1 \leq x \leq 5\) is \(m\).
The graph of \(y = \mathbf{g}(x)\) is a reflection in the \(x\)-axis of \(y = \mathbf{f}(x)\).
The graph of \(y = \mathbf{h}(x)\) is a translation of \(y = \mathbf{g}(x)\) by \(\begin{bmatrix} 3 \\ 7 \end{bmatrix}\)
Determine, in terms of \(m\), the mean value of the function \(\mathbf{h}\) over the interval \(4 \leq x \leq 8\)
[2 marks]
Selects a method to determine the mean value of by describing the
effect of either transformation on the graph of .
PI by -m or km 7. ℎ
Answer
Marks
Guidance
𝑦𝑦 = 𝑓𝑓(𝑥𝑥)
3.1a
M1
= −𝑚𝑚
mean
±
Answer
Marks
Guidance
Obtains the correct answer .
1.1b
A1
7−𝑚𝑚
Answer
Marks
Guidance
Total
2
= −𝑚𝑚+7
Q
Marking instructions
AO
Question 12:
12 | Selects a method to determine the mean value of by describing the
effect of either transformation on the graph of .
PI by -m or km 7. ℎ
𝑦𝑦 = 𝑓𝑓(𝑥𝑥) | 3.1a | M1 | mean
= −𝑚𝑚
mean
±
Obtains the correct answer . | 1.1b | A1
7−𝑚𝑚
Total | 2 | = −𝑚𝑚+7
Q | Marking instructions | AO | Marks | Typical solution
The mean value of the function $\mathbf{f}$ over the interval $1 \leq x \leq 5$ is $m$.
The graph of $y = \mathbf{g}(x)$ is a reflection in the $x$-axis of $y = \mathbf{f}(x)$.
The graph of $y = \mathbf{h}(x)$ is a translation of $y = \mathbf{g}(x)$ by $\begin{bmatrix} 3 \\ 7 \end{bmatrix}$
Determine, in terms of $m$, the mean value of the function $\mathbf{h}$ over the interval $4 \leq x \leq 8$
[2 marks]
\hfill \mbox{\textit{AQA Further AS Paper 1 2020 Q12 [2]}}