Translation with root finding

Questions where a cubic is given in factored form, expanded, then translated (horizontally or vertically), and students must find roots or factors of the transformed function.

12 questions

OCR MEI C1 2015 June Q10
10 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).
  1. Sketch the curve \(y = \mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
  3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
  4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 Q1
1 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).
  1. Sketch the curve \(y = \mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
  3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
  4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 Q3
3 You are given that \(\mathrm { f } ( x ) = ( 2 x - 3 ) ( x + 2 ) ( x + 4 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. State the roots of \(\mathrm { f } ( x - 2 ) = 0\).
  3. You are also given that \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + 15\).
    (A) Show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 2 x - 9\).
    (B) Show that \(\mathrm { g } ( 1 ) = 0\) and hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 Q5
5 A cubic curve has equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis where \(x = - , \frac { 1 } { 2 }\) and 5 .
  1. Write down three linear factors of \(\mathrm { f } ( x )\). Hence find the equation of the curve in the form \(y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 0 } { - 8 }\). State the coordinates of the point where the translated curve intersects the \(y\)-axis.
  4. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression in factorised form for \(\mathrm { g } ( x )\) and state the coordinates of the point where the curve \(y = \mathrm { g } ( x )\) intersects the \(y\)-axis.
OCR MEI C1 Q3
3
  1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
    (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
    (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
    Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
  2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.
OCR MEI C1 Q3
3 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).
  1. Sketch the curve \(y = \mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
  3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
  4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 Q4
4
  1. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
    (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
  2. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
    (A) Show that \(\mathrm { g } ( 5 ) = 0\).
    (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
    (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
  3. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).
OCR MEI C1 2011 January Q12
12
  1. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
    (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
  2. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
    (A) Show that \(\mathrm { g } ( 5 ) = 0\).
    (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
    (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
  3. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).
OCR MEI C1 2009 June Q12
12
  1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
    (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
    (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
    Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
  2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.
OCR MEI C1 2013 June Q11
11 You are given that \(\mathrm { f } ( x ) = ( 2 x - 3 ) ( x + 2 ) ( x + 4 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. State the roots of \(\mathrm { f } ( x - 2 ) = 0\).
  3. You are also given that \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + 15\).
    (A) Show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 2 x - 9\).
    (B) Show that \(\mathrm { g } ( 1 ) = 0\) and hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 Q11
11 A cubic polynomial is given by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8\).
  1. Show that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\). Factorise \(\mathrm { f } ( x )\) fully.
    Sketch the graph of \(y = f ( x )\).
  2. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { - 3 } { 0 }\). Write down an equation for the resulting graph. You need not simplify your answer.
    Find also the intercept on the \(y\)-axis of the resulting graph.
SPS SPS SM Pure 2022 June Q8
4 marks
8. The function \(\mathrm { f } ( x )\) is such that \(\mathrm { f } ( x ) = - x ^ { 3 } + 2 x ^ { 2 } + k x - 10\) The graph of \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis at the points with coordinates \(( a , 0 ) , ( 2,0 )\) and \(( b , 0 )\) where \(a < b\)
  1. Show that \(k = 5\)
    [0pt] [1 mark]
  2. Find the exact value of \(a\) and the exact value of \(b\)
    [0pt] [3 marks]
  3. The functions \(\mathrm { g } ( x )\) and \(\mathrm { h } ( x )\) are such that $$\begin{aligned} & g ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 10
    & h ( x ) = - 8 x ^ { 3 } + 8 x ^ { 2 } + 10 x - 10 \end{aligned}$$
    1. Explain how the graph of \(y = \mathrm { f } ( x )\) can be transformed into the graph of \(y = \mathrm { g } ( x )\) Fully justify your answer.
  5. (ii) Explain how the graph of \(y = \mathrm { f } ( x )\) can be transformed into the graph of \(y = \mathrm { h } ( x )\) Fully justify your answer.
    [0pt] [BLANK PAGE] A geometric series has second term 16 and fourth term 8 All the terms of the series are positive. The \(n\)th term of the series is \(u _ { n }\)
    Find the exact value of \(\sum _ { n = 5 } ^ { \infty } u _ { n }\)
    [0pt] [BLANK PAGE]