Transformations of Graphs Flashcards

All 11 cards in this deck

What transformation maps the graph of $y=f(x)$ onto the graph of $y=f(x)+a$?

e.g. $y=f(x)-4$

A translation by the vector $\begin{pmatrix}0\\a\end{pmatrix}$ ($a$ units in the positive $y$-direction).

e.g. $y=f(x)-4$ is a translation 4 units down.

What transformation maps the graph of $y=f(x)$ onto the graph of $y=f(x+a)$?

e.g. $y=f(x+2)$

A translation by the vector $\begin{pmatrix}-a\\0\end{pmatrix}$ ($a$ units in the negative $x$-direction).

e.g. $y=f(x+2)$ is a translation 2 units in the negative $x$-direction.

True or false? The graph of $y=f(x-3)$ is the graph of $y=f(x)$ translated 3 units to the left.

False. It is translated 3 units to the right, i.e. by $\begin{pmatrix}3\\0\end{pmatrix}$.

What are the steps to draw a combined translation of a given graph of $y=f(x)$?

e.g. draw $y=f(x+3)-1$

1. Change inside the bracket $\to$ horizontal move: $+3$ gives 3 left.
2. Change outside the bracket $\to$ vertical move: $-1$ gives 1 down.
3. Translate every key point by that vector: $\begin{pmatrix}-3\\-1\end{pmatrix}$.

The turning point of $y=g(x)$ has coordinates $(-3,6)$. What are the coordinates of the turning point of $y=g(x-7)$?

$(4,6)$.
The graph moves 7 units in the positive $x$-direction, so only the $x$-coordinate changes.

The graph of $y=\sin x^\circ$ is translated 90° in the negative $x$-direction. Write down an equation of the new graph.

$y=\sin(x+90)^\circ$.

What transformation maps the graph of $y=f(x)$ onto the graph of $y=-f(x)$?

A reflection in the $x$-axis.
Each point $(p,q)$ maps to $(p,-q)$.

What transformation maps the graph of $y=f(x)$ onto the graph of $y=f(-x)$?

A reflection in the $y$-axis.
Each point $(p,q)$ maps to $(-p,q)$.

True or false? The graph of $y=f(-x)$ is the graph of $y=f(x)$ reflected in the $x$-axis.

False. It is a reflection in the $y$-axis; $y=-f(x)$ is the reflection in the $x$-axis.

How do you find where a point on $y=f(x)$ moves to on a transformed curve?

e.g. $(7,2)$ on $y=f(x)$, mapped onto $y=f(-x)-3$

Apply each change to the coordinates in turn: $f(-x)$ reflects in the $y$-axis (change the sign of $x$); $-3$ outside the bracket moves 3 down (subtract 3 from $y$).
$(7,2) \to (-7,2) \to (-7,-1)$

The graph of $y=\sin x^\circ$ is reflected in the $x$-axis. Write down an equation of the reflected graph.

$y=-\sin x^\circ$ (or equivalently $y=\sin(-x)^\circ$).