Combinations of Transformations (AQA A-Level Mathematics): Revision Notes

Examples: Describe in words the following transformations applied to $f(x)$.

1. Translation $f(x) \longmapsto f(x + 2)$

• Translation by $(-2, 0)$
• Moves the graph 2 units to the left.

2. Translation $f(x) \longmapsto f(x) + 6$

• Translation by $(0, 6)$
• Moves the graph 6 units up.

3. Reflection $f(x) \longmapsto f(-x)$

• Reflect in the y-axis (i.e., reflection in the $x$-direction).

4. Stretching $f(x) \longmapsto 3f(x)$

• Stretch by a scale factor of 3 parallel to the $y$-axis.

5. Stretching $f(x) \longmapsto f(6x)$

• Stretch by a scale factor of $\frac{1}{6}$ parallel to the $x$-axis.

Example: Translate $y = x^2$ by $(0, 4)$

• $y = x^2 \longmapsto y = x^2 + 4$.
• Original Function: $y = x^2$
• Transformation: Translate up by $4$ units
• New Function: $y = x^2 + 4$

Example: Reflect $y = x^3$ in the $y$-axis

• $y = x^3 \longmapsto y = (-x)^3$.
• Original Function: $y = x^3$
• Transformation: Reflect in the $y$-axis
• New Function: $y = (-x)^3$

Example: Reflect $y = x^3$ in the $x$-axis

• $y = x^3 \longmapsto y = -x^3$.
• Original Function: $y = x^3$
• Transformation: Reflect in the $x$-axis
• New Function: $y = -x^3$

Graph Examples: The following graph shows $y=f(x)$

1. Sketching $y = 2f(x)$

• Stretch by scale factor 2 parallel to the $y$-axis.

2. Translation $y = f(x + 3)$

• Translate by $(-3, 0)$.

Find the Equation of the Curve when the graph of $y = x^2 + 2x + 1$ is transformed as follows

Summary

• Translations: Move the graph up, down, left, or right.
• Stretches: Change the shape of the graph by scaling it.
• Reflections: Flip the graph over a specified axis.

Example 1: Describe fully the following transformation:

6. Translate $\binom{0}{3}$ then Stretch S.F. $\frac{1}{2}$ parallel to the $x$-axis.

• OR-

7. Stretch S.F. $\frac{1}{2}$ parallel to the $x$-axis then Translate $\binom{0}{3}$.

Notice: In the above example, the order of transformations being performed doesn't matter. Since one is in the x-direction and one is in the y-direction, they are independent of each other.

Example 2: Describe fully the transformation that transforms

Note: Order is important as both transformations are in the $y$-direction.

8. First, $\times 5$: Stretch by S.F. 5 parallel to the $y$-axis.
9. Next, $+3$: Translate $\binom{0}{3}$.

Example: Describe fully the transformations that transform

Note here there are 3 separate transformations.

Two are "inside," and for "inside" transformations, the usual order is reversed.

3. Stretch by S.F. 3 parallel to $y$-axis. $y$ independent of the two $x$ transformations so can be placed anywhere in this example.

4. Translate $\binom{-3}{0}$

5. Stretch by S.F. $\frac{1}{2}$ parallel to $x$-axis. $x$ transformations is opposite order to BIDMAS.

Question 1 (June 2005, Q9i [Modified])

The function $f$ is defined by $f(x) = \sqrt{mx+7} - 4$, where $x \geq -\frac{7}{m}$ and $m$ is a positive constant.

(i) A sequence of transformations maps the curve $y = \sqrt{x}$ to the curve $y = f(x)$. Give details of these transformations.

10. Translate $\binom{-7}{0}$
11. Stretch by S.F. $\frac{1}{m}$ parallel to $x$-axis.
12. Translate $\binom{0}{-4}$