Single transformation application

3

1. The curve \(y = 5 \sqrt { } x\) is transformed by a stretch, scale factor \(\frac { 1 } { 2 }\), parallel to the \(x\)-axis. Find the equation of the curve after it has been transformed.
2. Describe the single transformation which transforms the curve \(y = 5 \sqrt { } x\) to the curve \(y = ( 5 \sqrt { } x ) - 3\).

2

1. Sketch the curve \(y = - \frac { 1 } { x }\).
2. The curve \(y = - \frac { 1 } { x }\) is translated by 2 units parallel to the \(x\)-axis in the positive direction. State the equation of the transformed curve.
3. Describe a transformation that transforms the curve \(y = - \frac { 1 } { x }\) to the curve \(y = - \frac { 1 } { 3 x }\).

3

1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
2. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is translated by 2 units in the positive \(x\)-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 \(\frac { 1 } { 2 }\) and, as a result, the point \(\left( \frac { 1 } { 2 } , - 4 \right)\) on the curve is transformed to the point \(P\). State the coordinates of \(P\).

9 The curve \(y = ( x - 1 ) ^ { 2 }\) maps onto the curve \(\mathrm { C } _ { 1 }\) following a stretch scale factor \(\frac { 1 } { 2 }\) in the \(x\)-direction.

1. Show that the equation of \(\mathrm { C } _ { 1 }\) can be written as \(y = 4 x ^ { 2 } - 4 x + 1\). The curve \(\mathrm { C } _ { 2 }\) is a translation of \(y = 4.25 x - x ^ { 2 }\) by \(\binom { 0 } { - 3 }\).
2. Show that the normal to the curve \(\mathrm { C } _ { 1 }\) at the point \(( 0,1 )\) is a tangent to the curve \(\mathrm { C } _ { 2 }\).

Curve C has equation \(y = x^2\) C is translated by vector \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give curve \(C_1\) Line L has equation \(y = x\) L is stretched by scale factor 2 parallel to the \(x\)-axis to give line \(L_1\) Find the exact distance between the two intersection points of \(C_1\) and \(L_1\) [6 marks]

Curve \(C\) has equation \(y = 8 \sin x\)

1. Curve \(C\) is transformed onto curve \(C_1\) by a translation of vector \(\begin{pmatrix} 0 \\ 4 \end{pmatrix}\) Find the equation of \(C_1\) [1 mark]
2. Curve \(C\) is transformed onto curve \(C_2\) by a stretch of scale factor 4 in the \(y\) direction. Find the equation of \(C_2\) [1 mark]
3. Curve \(C\) is transformed onto curve \(C_3\) by a stretch of scale factor 2 in the \(x\) direction. Find the equation of \(C_3\) [1 mark]