Sketch absolute value of function

Questions asking to sketch y = |f(x)| or related absolute value transformations, given the original curve or its equation.

3 questions

Edexcel C3 2007 January Q7
7. $$f ( x ) = x ^ { 4 } - 4 x - 8$$
  1. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \([ - 2 , - 1 ]\).
  2. Find the coordinates of the turning point on the graph of \(y = \mathrm { f } ( x )\).
  3. Given that \(\mathrm { f } ( x ) = ( x - 2 ) \left( x ^ { 3 } + a x ^ { 2 } + b x + c \right)\), find the values of the constants, \(a , b\) and \(c\).
  4. In the space provided on page 21, sketch the graph of \(y = \mathrm { f } ( x )\).
  5. Hence sketch the graph of \(y = | \mathrm { f } ( x ) |\).
OCR C3 Q9
9.
\includegraphics[max width=\textwidth, alt={}, center]{5e6a37a1-c51f-4637-aaae-48da6ab3eca0-3_727_1022_244_342} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the axes at \(( p , 0 )\) and \(( 0 , q )\) and the lines \(x = 1\) and \(y = 2\) are asymptotes of the curve.
  1. Showing the coordinates of any points of intersection with the axes and the equations of any asymptotes, sketch on separate diagrams the graphs of
    1. \(y = | \mathrm { f } ( x ) |\),
    2. \(y = 2 \mathrm { f } ( x + 1 )\). Given also that $$\mathrm { f } ( x ) \equiv \frac { 2 x - 1 } { x - 1 } , \quad x \in \mathbb { R } , \quad x \neq 1$$
  2. find the values of \(p\) and \(q\),
  3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
AQA Further Paper 1 2021 June Q7
7 The diagram below shows the graph of \(y = \mathrm { f } ( x ) \quad ( - 4 \leq x \leq 4 )\) The graph meets the \(x\)-axis at \(x = 1\) and \(x = 3\)
The graph meets the \(y\)-axis at \(y = 2\)
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-10_846_854_539_593} 7
  1. Sketch the graph of \(y = | \mathrm { f } ( x ) |\) on the axes below.
    Show any axis intercepts.
    \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-10_844_844_1601_598} 7
  2. Sketch the graph of \(y = \frac { 1 } { \mathrm { f } ( x ) }\) on the axes below.
    Show any axis intercepts and asymptotes.
    \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-11_844_846_495_603} 7
  3. Sketch the graph of \(y = \mathrm { f } ( | x | )\) on the axes below.
    Show any axis intercepts.
    \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-11_848_849_1647_593}