Vertical translation of cubic with factorisation

Questions where a cubic f(x) is given and a vertical translation g(x) = f(x) + k is applied, requiring students to expand, verify a root, and fully factorise the new cubic g(x).

8 questions · Moderate -0.7

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem 1.02n Sketch curves: simple equations including polynomials 1.02w Graph transformations: simple transformations of f(x)

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OCR MEI C1 2015 June Q10

12 marks Moderate -0.8

10 You are given that $\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )$.

Sketch the curve $y = \mathrm { f } ( x )$.
Show that $\mathrm { f } ( x )$ may be written as $x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30$.
Describe fully the transformation that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { g } ( x )$, where $\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6$.
Show that $\mathrm { g } ( - 1 ) = 0$. Hence factorise $\mathrm { g } ( x )$ completely.

OCR MEI C1 Q1

12 marks Moderate -0.8

1 You are given that $\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )$.

Sketch the curve $y = \mathrm { f } ( x )$.
Show that $\mathrm { f } ( x )$ may be written as $x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30$.
Describe fully the transformation that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { g } ( x )$, where $\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6$.
Show that $\mathrm { g } ( - 1 ) = 0$. Hence factorise $\mathrm { g } ( x )$ completely.

OCR MEI C1 Q3

12 marks Moderate -0.8

3 You are given that $\mathrm { f } ( x ) = ( 2 x - 3 ) ( x + 2 ) ( x + 4 )$.

Sketch the graph of $y = \mathrm { f } ( x )$.
State the roots of $\mathrm { f } ( x - 2 ) = 0$.
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 Q3

12 marks Moderate -0.8

3 You are given that $\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )$.

Sketch the curve $y = \mathrm { f } ( x )$.
Show that $\mathrm { f } ( x )$ may be written as $x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30$.
Describe fully the transformation that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { g } ( x )$, where $\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6$.
Show that $\mathrm { g } ( - 1 ) = 0$. Hence factorise $\mathrm { g } ( x )$ completely.

OCR MEI C1 Q4

13 marks Moderate -0.8

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 $\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20$.
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.
Describe fully the transformation that maps $y = \mathrm { f } ( x )$ onto $y = \mathrm { g } ( x )$.

OCR MEI C1 2011 January Q12

13 marks Moderate -0.8

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 $\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20$.
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.
Describe fully the transformation that maps $y = \mathrm { f } ( x )$ onto $y = \mathrm { g } ( x )$.

OCR MEI C1 2013 June Q11

12 marks Moderate -0.8

You are given that $\text{f}(x) = (2x - 3)(x + 2)(x + 4)$.

Sketch the graph of $y = \text{f}(x)$. [3]
State the roots of $\text{f}(x - 2) = 0$. [2]
You are also given that $\text{g}(x) = \text{f}(x) + 15$.
1. Show that $\text{g}(x) = 2x^3 + 9x^2 - 2x - 9$. [2]
2. Show that $\text{g}(1) = 0$ and hence factorise $\text{g}(x)$ completely. [5]

AQA Paper 3 2023 June Q6

9 marks Standard +0.3

Sketch the curve with equation $$y = x^2(2x + a)$$ where $a > 0$ [3 marks] \includegraphics{figure_6a}
The polynomial $p(x)$ is given by $$p(x) = x^2(2x + a) + 36$$
1. It is given that $x + 3$ is a factor of $p(x)$ Use the factor theorem to show $a = 2$ [2 marks]
2. State the transformation which maps the curve with equation $$y = x^2(2x + 2)$$ onto the curve with equation $$y = x^2(2x + 2) + 36$$ [2 marks]
3. The polynomial $x^2(2x + 2) + 36$ can be written as $(x + 3)(2x^2 + bx + c)$ Without finding the values of $b$ and $c$, use your answers to parts (a) and (b)(ii) to explain why $$b^2 < 8c$$ [2 marks]