Single transformation application

2

1. The curve \(y = x ^ { 2 }\) is translated 2 units in the positive \(x\)-direction. Find the equation of the curve after it has been translated.
2. The curve \(y = x ^ { 3 } - 4\) is reflected in the \(x\)-axis. Find the equation of the curve after it has been reflected.

4

1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
2. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is translated by 3 units in the negative \(x\)-direction. State the equation of the curve after it has been translated.
3. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 4 and, as a result, the point \(P ( 1,1 )\) is transformed to the point \(Q\). State the coordinates of \(Q\).

5

1. Sketch the curve \(y = - x ^ { 3 }\).
2. The curve \(y = - x ^ { 3 }\) is translated by 3 units in the positive \(x\)-direction. Find the equation of the curve after it has been translated.
3. Describe a transformation that transforms the curve \(y = - x ^ { 3 }\) to the curve \(y = - 5 x ^ { 3 }\).

2

1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
2. Sketch the curve \(y = 3 - \frac { 1 } { x ^ { 2 } }\).
3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 2 . State the equation of the transformed curve.

1 Find the values of \(P , Q , R\) and \(S\) in the identity \(3 x ^ { 3 } + 18 x ^ { 2 } + P x + 31 \equiv Q ( x + R ) ^ { 3 } + S\).

3 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + p x + q\), where \(p\) and \(q\) are constants.

1. 1. Given that \(\mathrm { f } ^ { \prime } ( 2 ) = 13\), find the value of \(p\).
   2. Given also that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(q\). The curve \(y = \mathrm { f } ( x )\) is translated by the vector \(\binom { 2 } { - 3 }\).
2. Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are integers to be found.

1 The equation of a curve is \(y = 4 x ^ { 2 } + 8 x + 1\).
The curve is stretched parallel to the \(x\)-axis with scale factor 2 .
Find the equation of the new curve, giving your answer in the form \(\mathrm { y } = a \mathrm { x } ^ { 2 } + b \mathrm { x } + c\), where \(a , b\) and \(c\) are integers to be determined.

2 The curve \(y = x ^ { 3 } - 2 x\) is translated by the vector \(\binom { 1 } { - 4 }\). Write down the equation of the translated curve. [2]

6

1. Sketch the curve with equation $$y = x ^ { 2 } ( 2 x + a )$$ where \(a > 0\)
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2. The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 2 } ( 2 x + a ) + 36$$ 6
3. 1. It is given that \(x + 3\) is a factor of \(\mathrm { p } ( x )\)
      Use the factor theorem to show \(a = 2\) 6
4. (ii) State the transformation which maps the curve with equation $$y = x ^ { 2 } ( 2 x + 2 )$$ onto the curve with equation $$y = x ^ { 2 } ( 2 x + 2 ) + 36$$ 6
5. (iii) The polynomial \(x ^ { 2 } ( 2 x + 2 ) + 36\) can be written as \(( x + 3 ) \left( 2 x ^ { 2 } + b x + c \right)\)
   Without finding the values of \(b\) and \(c\), use your answers to parts (a) and (b)(ii) to explain why $$b ^ { 2 } < 8 c$$