Transformations of graphs

Algebra

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The graph of $y = x^{2}$ is translated by the vector $\begin{pmatrix} 0 \\ 3 \end{pmatrix}$. Write the equation of the new graph.

2 marks
The graph of $y = x^{2}$ is translated by $\begin{pmatrix} 2 \\ 0 \end{pmatrix}$. Write the new equation.

2 marks
Describe the transformation that maps $y = x^{2}$ to $y = (x + 1)^{2}$.

3 marks
Describe the transformations that map $y = f(x)$ to $y = f(x) + 2$ and to $y = f(x - 3)$.

3 marks
Describe the transformation from $y = \sin x$ to $y = \sin x + 1$, and from $y = \sin x$ to $y = \sin(x - 30°)$.

4 marks
Describe the transformation from $y = x^{2}$ to $y = 2x^{2}$, and from $y = x^{2}$ to $y = (2x)^{2}$.

4 marks
The graph of $y = f(x)$ has a turning point at $(2, 5)$. Write the turning points of $y = f(x) + 4$, $y = f(x - 3)$, $y = f(x + 1) - 2$ and $y = -f(x)$.

5 marks
Write $y = x^{2} - 4x + 1$ in completed-square form. Describe the translation that maps $y = x^{2}$ onto this curve. The curve is then reflected in the $x$-axis. Write the equation of the image.

6 marks

Mark scheme

Answer. $y = x^{2} + 3$
- The translation is $3$ units up.
- $y = x^{2} + 3$
Answer. $y = (x - 2)^{2}$
- A move of $2$ to the right replaces $x$ by $x - 2$.
- $y = (x - 2)^{2}$
Answer. A translation by $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$
- The bracket is $x - (-1)$.
- The graph moves $1$ unit left.
Answer. Up $2$, and right $3$
- Adding $2$ outside the function moves the graph up $2$.
- $f(x - 3)$ moves the graph right $3$.
Answer. A translation up $1$, and a translation right by $30°$
- $+ 1$ is a vertical translation.
- $(x - 30°)$ is a horizontal translation of $30°$ to the right.
Answer. A stretch of scale factor $2$ parallel to the $y$-axis, and a stretch of scale factor $\dfrac{1}{2}$ parallel to the $x$-axis.
- Multiplying the whole function by $2$ stretches vertically.
- Replacing $x$ by $2x$ stretches horizontally with scale factor $\dfrac{1}{2}$.
Answer. $(2, 9)$, $(5, 5)$, $(1, 3)$ and $(2, -5)$
- Up $4$ gives $(2, 9)$.
- Right $3$ gives $(5, 5)$.
- Left $1$ and down $2$ gives $(1, 3)$.
- Reflection in the $x$-axis gives $(2, -5)$.
Answer. $(x - 2)^{2} - 3$. Translation by $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$. Image $y = -x^{2} + 4x - 1$.
- $y = (x - 2)^{2} - 4 + 1 = (x - 2)^{2} - 3$.
- This is $y = x^{2}$ moved by $(2, -3)$.
- Reflection in the $x$-axis multiplies the original expression by $-1$.
- $y = -(x^{2} - 4x + 1) = -x^{2} + 4x - 1$.