Transformations (AQA A-Level Further Maths): Revision Notes

Worked Example: Enlarging an Ellipse

Problem: An ellipse with equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$ is enlarged by scale factor 3.

(a) Find the equation of the transformed curve and state its points of intersection with the axes.

(b) Sketch the transformed curve.

Solution:

(a) Apply the enlargement rule by substituting $x = \frac{x}{3}$ and $y = \frac{y}{3}$:

$\left(\frac{x}{3}\right)^2/4 + \left(\frac{y}{3}\right)^2/9 = 1$

Simplifying: $\frac{x^2}{36} + \frac{y^2}{81} = 1$

To find the intersection points with the axes:

• When $x = 0$: $\frac{y^2}{81} = 1 \Rightarrow y^2 = 81 \Rightarrow y = \pm 9$

Coordinates: (0, 9) and (0, -9)

• When $y = 0$: $\frac{x^2}{36} = 1 \Rightarrow x^2 = 36 \Rightarrow x = \pm 6$

Coordinates: (6, 0) and (-6, 0)

(b) Sketch the curve using the coordinates found in part (a):

Worked Example: Rotating a Hyperbola

Problem: The hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$ is rotated anticlockwise through $\frac{3\pi}{2}$ radians about $(0, 0)$.

(a) Write down the equation of the transformed curve.

(b) Sketch the transformed curve and find the equations of the asymptotes.

Solution:

(a) For a rotation of $\frac{3\pi}{2}$ radians, replace $x$ by $-y$ and $y$ by $x$:

$\frac{(-y)^2}{9} - \frac{(x)^2}{16} = 1$

Simplifying: $\frac{y^2}{9} - \frac{x^2}{16} = 1$

(b) For a hyperbola of the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the asymptotes are $y = \pm\frac{b}{a}x$.

The original hyperbola has asymptotes: $y = \pm\frac{4}{3}x$

After the rotation, the transformed curve has asymptotes: $y = \pm\frac{3}{4}x$

We can verify this by applying the same transformation to the asymptote equations.

Worked Example: Multiple Transformations

Problem: The ellipse with equation $x^2 + \frac{y^2}{4} = 1$ is enlarged by scale factor 2, then rotated by $\frac{\pi}{2}$ radians anticlockwise about $(0, 0)$ before being translated by vector $\begin{pmatrix} 3 \ 2 \end{pmatrix}$.

Find the equation of the new conic in the form $ax^2 + by^2 + cx + dy + e = 0$ where $a, b, c, d$ and $e$ are constants, and sketch the graph.

Solution:

Step 1: Enlargement (scale factor 2)

Replace $x$ with $\frac{x}{2}$ and $y$ with $\frac{y}{2}$:

$\left(\frac{x}{2}\right)^2 + \frac{\left(\frac{y}{2}\right)^2}{4} = 1$

Simplifying: $\frac{x^2}{4} + \frac{y^2}{16} = 1$

Step 2: Rotation ($\frac{\pi}{2}$ radians)

Replace $x$ by $y$ and $y$ by $-x$:

$\frac{(y)^2}{4} + \frac{(-x)^2}{16} = 1$

Simplifying: $\frac{y^2}{4} + \frac{x^2}{16} = 1$

Or equivalently: $\frac{x^2}{16} + \frac{y^2}{4} = 1$

Step 3: Translation by $\begin{pmatrix} 3 \ 2 \end{pmatrix}$

Replace $x$ with $(x - 3)$ and $y$ with $(y - 2)$:

$\frac{(x-3)^2}{16} + \frac{(y-2)^2}{4} = 1$

Step 4: Expand to standard form

Multiply through by 16: $(x-3)^2 + 4(y-2)^2 = 16$

Expanding: $x^2 - 6x + 9 + 4y^2 - 16y + 16 = 16$

Simplifying: $x^2 + 4y^2 - 6x - 16y + 9 = 0$

Worked Example: Identifying Transformations of a Parabola

Problem: A parabola $C$ has equation $y^2 = x$.

Describe a sequence of two transformations which maps $C$ onto the curve with equation $y^2 - 6y + x + 11 = 0$.

Solution:

Step 1: Rearrange into standard form

Starting with: $y^2 - 6y + x + 11 = 0$

Complete the square for $y$:

Step 2: Identify transformations

Comparing $(y - 3)^2 = -(x + 2)$ with the original $y^2 = x$:

• The term (y - 3)² indicates a translation of 3 units in the positive $y$-direction.
• The term -x indicates a reflection in the $y$-axis.
• The term (x + 2) indicates a translation of 2 units in the positive $x$-direction (since $x$ is replaced by $x + 2$, this means moving 2 units left, but combined with the negative sign, it's 2 units right after reflection).

Step 3: Determine the order

One possible sequence is: translation using vector $\begin{pmatrix} 2 \ 3 \end{pmatrix}$, followed by a reflection in the $y$-axis.

Alternative interpretation:

The equation can also be written as $(y - 3)^2 = -(x + 2)$.

This represents a reflection in the $y$-axis followed by a translation using vector $\begin{pmatrix} -2 \ 3 \end{pmatrix}$.

Key insight: The order of transformations matters. Different sequences can produce the same final result, but you must specify the order clearly.

Worked Example: Identifying Hyperbola Transformations

Problem: A hyperbola $H$ has equation $x^2 - \frac{y^2}{3} = 1$.

Describe a sequence of transformations which maps $H$ onto the curve with equation $12x^2 = y^2 + 48x - 36$.

Solution:

Step 1: Rearrange into standard form

Starting with: $12x^2 = y^2 + 48x - 36$

Rearranging:

Complete the square for $x$:

This can be written as: $\frac{(x-2)^2}{1} - \frac{y^2}{12} = 1$

Or: $(x-2)^2 - \frac{\left(\frac{y}{2}\right)^2}{3} = 1$

Step 2: Identify transformations

Comparing with the original $x^2 - \frac{y^2}{3} = 1$:

• The term (x - 2)² indicates a translation of 2 units in the positive $x$-direction.
• The term (y/2)² indicates a stretch parallel to the $y$-axis with scale factor 2.

Step 3: Determine the order

A possible sequence is: a stretch parallel to the y-axis with scale factor 2, followed by a translation in the positive x-direction of 2 units.

Note: These transformations can be applied in either order in this case because the stretch is parallel to the $y$-axis and the translation is in the $x$-direction, so they act independently on different variables.