Horizontal translation of factored polynomial

Questions where a factored or expanded polynomial is sketched, then a horizontal translation is applied by replacing x with (x - a), requiring students to find new roots or describe the transformation.

8 questions · Moderate -0.5

1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x)
Sort by: Default | Easiest first | Hardest first
Edexcel C1 2009 June Q10
9 marks Moderate -0.8
10. (a) Factorise completely \(x ^ { 3 } - 6 x ^ { 2 } + 9 x\) (b) Sketch the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. Using your answer to part (b), or otherwise,
(c) sketch, on a separate diagram, the curve with equation $$y = ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 9 ( x - 2 )$$ showing the coordinates of the points at which the curve meets the \(x\)-axis.
Edexcel C1 Q7
9 marks Moderate -0.8
7. (a) Factorise completely \(x ^ { 3 } - 4 x\).
(3)
(b) Sketch the curve with equation \(y = x ^ { 3 } - 4 x\), showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
(c) On a separate diagram, sketch the curve with equation \(y = ( x - 1 ) ^ { 3 } - 4 ( x - 1 ) ,\) showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
\end{tabular} & Leave blank
\hline \end{tabular} \end{center}
\includegraphics[max width=\textwidth, alt={}]{6400bb0c-f199-45f2-a4b1-55534e2c63b0-11_2608_1924_141_75}
\begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
OCR MEI C1 Q4
5 marks Moderate -0.3
4 You are given that \(\mathrm { f } ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. State the values of \(x\) which satisfy \(\mathrm { f } ( x + 3 ) = 0\).
OCR MEI C1 2013 January Q9
5 marks Moderate -0.8
9 You are given that \(\mathrm { f } ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. State the values of \(x\) which satisfy \(\mathrm { f } ( x + 3 ) = 0\).
OCR C1 2013 January Q3
5 marks Moderate -0.3
  1. Sketch the curve \(y = (1 + x)(2 - x)(3 + x)\), giving the coordinates of all points of intersection with the axes. [3]
  2. Describe the transformation that transforms the curve \(y = (1 + x)(2 - x)(3 + x)\) to the curve \(y = (1 - x)(2 + x)(3 - x)\). [2]
OCR MEI C1 2009 June Q12
13 marks Moderate -0.8
  1. You are given that \(\text{f}(x) = (x + 1)(x - 2)(x - 4)\).
    1. Show that \(\text{f}(x) = x^3 - 5x^2 + 2x + 8\). [2]
    2. Sketch the graph of \(y = \text{f}(x)\). [3]
    3. The graph of \(y = \text{f}(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\). 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. [3]
  2. Show that 3 is a root of \(x^3 - 5x^2 + 2x + 8 = -4\). Hence solve this equation completely, giving the other roots in surd form. [5]
OCR MEI C1 2012 June Q11
12 marks Moderate -0.3
A cubic curve has equation \(y = f(x)\). The curve crosses the \(x\)-axis where \(x = -\frac{1}{2}\), \(-2\) and \(5\).
  1. Write down three linear factors of \(f(x)\). Hence find the equation of the curve in the form \(y = 2x^3 + ax^2 + bx + c\). [4]
  2. Sketch the graph of \(y = f(x)\). [3]
  3. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 0 \\ -8 \end{pmatrix}\). State the coordinates of the point where the translated curve intersects the \(y\)-axis. [1]
  4. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give the curve \(y = g(x)\). Find an expression in factorised form for \(g(x)\) and state the coordinates of the point where the curve \(y = g(x)\) intersects the \(y\)-axis. [4]
Edexcel AS Paper 1 Specimen Q13
7 marks Standard +0.3
  1. Factorise completely \(x^3 + 10x^2 + 25x\) [2]
  2. Sketch the curve with equation $$y = x^3 + 10x^2 + 25x$$ showing the coordinates of the points at which the curve cuts or touches the \(x\)-axis. [2]
The point with coordinates \((-3, 0)\) lies on the curve with equation $$y = (x + a)^3 + 10(x + a)^2 + 25(x + a)$$ where \(a\) is a constant.
  1. Find the two possible values of \(a\). [3]