True or False? A translation can change the shape of a graph.

False. A translation only moves the graph: its shape, size and orientation are all left unchanged. That is exactly what separates a translation from a stretch or a reflection.

What happens to a graph's asymptotes under a translation?

They are translated along with the graph. An asymptote parallel to the direction of the translation is the exception, and stays exactly where it is.

What happens to a graph's asymptotes under a stretch?

They are stretched along with the graph. An asymptote that is a coordinate axis, or that is parallel to the direction of the stretch, is left unaffected.

What happens to a graph's asymptotes under a reflection?

They are reflected too, exactly as a straight line would be. An asymptote lying along a coordinate axis, or perpendicular to the axis being reflected in, is unaffected.

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Transformations of Functions (Edexcel A Level Maths: Pure): Flashcards

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Define a translation of a graph.

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Cards in this collection (17)

Define a translation of a graph.
A translation shifts a graph up, down, left or right in the $x y$ plane.
A particular translation is specified by a translation vector, giving how far horizontally and how far vertically.
$y = f (x) + a$ is a translation by $[\begin{matrix} 0 \\ a \end{matrix}]$ , and $y = f (x + a)$ is a translation by:
$[\begin{matrix} ______ \\ ______ \end{matrix}]$
$[\begin{matrix} - a \\ 0 \end{matrix}]$
The minus sign is what makes a positive $a$ move the graph to the left, which is the opposite of what most people expect.
Under $y = f (x) + a$ , which coordinates of a point change?
Only the $y$ -coordinates; every $x$ -coordinate stays exactly where it was.
For $y = f (x + a)$ it is the other way round, with the $y$ -coordinates unchanged.
True or False?
A translation can change the shape of a graph.
False.
A translation only moves the graph: its shape, size and orientation are all left unchanged.
That is exactly what separates a translation from a stretch or a reflection.
What happens to a graph's asymptotes under a translation?
They are translated along with the graph.
An asymptote parallel to the direction of the translation is the exception, and stays exactly where it is.
The graph of $y = \frac{1}{x}$ has an asymptote on each axis. Where are they after the translation to $y = \frac{1}{x - 2} + 3$ ?
They move with the graph, to $x = 2$ and $y = 3$ .
This is how a reciprocal graph can end up crossing an axis, which the untranslated version never does.
For any function, what do $y = a f (x)$ and $y = f (a x)$ do to the graph?
$y = a f (x)$ is a vertical stretch of scale factor $a$ , centred on the $x$ -axis.
$y = f (a x)$ is a horizontal stretch of scale factor $\frac{1}{a}$ , centred on the $y$ -axis.
True or False?
$y = f (3 x)$ is a horizontal stretch of scale factor $3$ .
False.
The scale factor is $\frac{1}{3}$ , so the graph is squashed towards the $y$ -axis rather than stretched away from it.
Which points do not move under a vertical stretch?
The points on the $x$ -axis, because their $y$ -coordinate is zero and multiplying zero by anything leaves it zero.
Under a horizontal stretch it is the points on the $y$ -axis that stay put.
Under $y = a f (x)$ with $a > 1$ , points move $______$ from the $x$ -axis; with $0 < a < 1$ they move $______$ it.
Under $y = a f (x)$ with $a > 1$ , points move away from the $x$ -axis; with $0 < a < 1$ they move towards it.
All of the movement is parallel to the $y$ -axis, so nothing shifts sideways.
What happens to a graph's asymptotes under a stretch?
They are stretched along with the graph.
An asymptote that is a coordinate axis, or that is parallel to the direction of the stretch, is left unaffected.
Why is the scale factor of $y = f (a x)$ equal to $\frac{1}{a}$ rather than $a$ ?
Because the graph reaches any given output at a smaller input: whatever $f$ did at $x$ , $f (a x)$ does at $\frac{x}{a}$ .
A larger $a$ therefore pulls the graph inwards rather than stretching it out.
$y = - f (x)$ is a reflection in the $______$ -axis, and $y = f (- x)$ is a reflection in the $______$ -axis.
$y = - f (x)$ is a reflection in the $x$ -axis, and $y = f (- x)$ is a reflection in the $y$ -axis.
The minus outside the function acts on the outputs; the minus inside acts on the inputs.
Which points stay where they are under a reflection?
The points on the axis of reflection.
For $y = - f (x)$ those are the points on the $x$ -axis, whose $y$ -coordinate is zero and so is unchanged by a sign flip.
Under $y = f (- x)$ , what happens to the coordinates of a point?
The $y$ -coordinates stay the same and the $x$ -coordinates have their signs flipped.
Every point not already on the $y$ -axis moves to the other side of it.
What happens to a graph's asymptotes under a reflection?
They are reflected too, exactly as a straight line would be.
An asymptote lying along a coordinate axis, or perpendicular to the axis being reflected in, is unaffected.
True or False?
A graph that is symmetrical about the $y$ -axis is unchanged by $y = f (- x)$ .
True.
If the graph already matches itself on both sides of the $y$ -axis, reflecting it in that axis maps it exactly onto itself.
$y = x^{2}$ and $y = \cos x$ both behave this way.

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