Horizontal translation of cubic with root finding

Questions where a cubic is given in factored form, then a horizontal translation f(x-a) or f(x+a) is applied, and students must find the new roots or write the equation of the translated graph.

3 questions · Moderate -0.6

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

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Edexcel P1 2019 June Q10

10 marks Moderate -0.8

A curve has equation $y = \mathrm { f } ( x )$, where

$$f ( x ) = ( x - 4 ) ( 2 x + 1 ) ^ { 2 }$$ The curve touches the $x$-axis at the point $P$ and crosses the $x$-axis at the point $Q$.

State the coordinates of the point $P$.
Find $f ^ { \prime } ( x )$.
Hence show that the equation of the tangent to the curve at the point where $x = \frac { 5 } { 2 }$ can be expressed in the form $y = k$, where $k$ is a constant to be found. The curve with equation $y = \mathrm { f } ( x + a )$, where $a$ is a constant, passes through the origin $O$.
State the possible values of $a$.
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OCR MEI C1 Q5

12 marks Moderate -0.8

5 A cubic curve has equation $y = \mathrm { f } ( x )$. The curve crosses the $x$-axis where $x = - , \frac { 1 } { 2 }$ and 5 .

Write down three linear factors of $\mathrm { f } ( x )$. Hence find the equation of the curve in the form $y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c$.
Sketch the graph of $y = \mathrm { f } ( x )$.
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.
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

13 marks Moderate -0.3

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.
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.