Edexcel C3 — Question 6

Exam BoardEdexcel
ModuleC3 (Core Mathematics 3)
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCurve Sketching
TypeSketch single transformation from given curve
DifficultyStandard +0.3 This is a standard C3 transformations question requiring sketches of horizontal translation and composition with |x|, plus solving a simple modulus equation. The transformations are routine textbook exercises, and part (d) involves straightforward case-by-case algebra. Slightly easier than average due to the mechanical nature of the tasks.
Spec1.02l Modulus function: notation, relations, equations and inequalities1.02w Graph transformations: simple transformations of f(x)
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-009_458_876_285_539}
\end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
  3. the value of \(a\) and the value of \(b\),
  4. the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).
Question 6:
(a) \(f(0.8) = 0.082\), \(f(0.9) = -0.089\) M1
Change of sign \(\Rightarrow\) root in \((0.8, 0.9)\) A1
(2)
(b) \(f'(x) = 2x - 3 - \sin\left(\frac{1}{2}x\right)\) M1 A1
Sets \(f'(x) = 0 \Rightarrow x = \frac{3 + \sin\left(\frac{1}{2}x_n\right)}{2}\) M1 A1*
(4)
(c) Sub \(x_0 = 2\) into \(x_{n+1} = \frac{3 + \sin\left(\frac{1}{2}x_n\right)}{2}\) M1
\(x_1 \approx 1.921\), \(x_2 \approx 1.91(0)\) and \(x_3 \approx 1.908\) A1 A1
(3)
(d) \([1.90775, 1.90785]\) M1
\(f'(1.90775) = -0.00016\ldots\) AND \(f'(1.90785) = 0.0000076\ldots\) M1
Change of sign \(\Rightarrow x = 1.9078\) A1
(3)
(12 marks)
Question 6:

(a) $f(0.8) = 0.082$, $f(0.9) = -0.089$ M1

Change of sign $\Rightarrow$ root in $(0.8, 0.9)$ A1

(2)

(b) $f'(x) = 2x - 3 - \sin\left(\frac{1}{2}x\right)$ M1 A1

Sets $f'(x) = 0 \Rightarrow x = \frac{3 + \sin\left(\frac{1}{2}x_n\right)}{2}$ M1 A1*

(4)

(c) Sub $x_0 = 2$ into $x_{n+1} = \frac{3 + \sin\left(\frac{1}{2}x_n\right)}{2}$ M1

$x_1 \approx 1.921$, $x_2 \approx 1.91(0)$ and $x_3 \approx 1.908$ A1 A1

(3)

(d) $[1.90775, 1.90785]$ M1

$f'(1.90775) = -0.00016\ldots$ AND $f'(1.90785) = 0.0000076\ldots$ M1

Change of sign $\Rightarrow x = 1.9078$ A1

(3)

(12 marks)
6.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
  \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-009_458_876_285_539}
\end{center}
\end{figure}

Figure 1 shows part of the graph of $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. The graph consists of two line segments that meet at the point $( 1 , a ) , a < 0$. One line meets the $x$-axis at $( 3,0 )$. The other line meets the $x$-axis at $( - 1,0 )$ and the $y$-axis at $( 0 , b ) , b < 0$.

In separate diagrams, sketch the graph with equation
\begin{enumerate}[label=(\alph*)]
\item $y = \mathrm { f } ( x + 1 )$,
\item $y = \mathrm { f } ( | x | )$.

Indicate clearly on each sketch the coordinates of any points of intersection with the axes.

Given that $\mathrm { f } ( x ) = | x - 1 | - 2$, find
\item the value of $a$ and the value of $b$,
\item the value of $x$ for which $\mathrm { f } ( x ) = 5 x$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel C3  Q6}}
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