True or False? Translating a graph maintains its shape, size, and orientation.

True. Translating a graph maintains its shape, size, and orientation. It only moves the graph up / down / left / right .

True or False? When a graph is translated, any asymptotes remain in the same location(s).

False. When a graph is translated, any asymptotes are also translated in the same way as the rest of the graph.

What are invariant points ?

Invariant points are points on a graph that do not change during a particular transformation.

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FrontTranslations of Graphs
True or False?
Translating a graph maintains its shape, size, and orientation.

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True or False?
Translating a graph maintains its shape, size, and orientation.
True.
Translating a graph maintains its shape, size, and orientation.
It only moves the graph up/down/left/right.
When a graph is translated by the vector $(\begin{matrix} a \\ b \end{matrix})$ , how many units does it move and in which direction(s)?
When a graph is translated by the vector $(\begin{matrix} a \\ b \end{matrix})$ , it moves $a$ units right and $b$ units upwards.
If $a$ is negative then it moves left, and if $b$ is negative it moves downwards.
A graph is transformed from $f (x)$ to $f (x - a)$ . Describe the transformation fully.
If a graph is transformed from $f (x)$ to $f (x - a)$ , this is a translation of $a$ units right.
A graph is transformed from $f (x)$ to $f (x) + b$ . Describe the transformation fully.
If a graph is transformed from $f (x)$ to $f (x) + b$ , it has been translated vertically upwards by $b$ units.
True or False?
When a graph is translated, any asymptotes remain in the same location(s).
False.
When a graph is translated, any asymptotes are also translated in the same way as the rest of the graph.
Outline how would you find a new equation for a graph which has been translated horizontally.
To find a new equation for a graph which has been translated horizontally, consider how the function has been changed.
It has changed from $f (x)$ to $f (x - a)$ , this is a translation of $a$ units right.
Therefore in the equation of the graph, you can simply replace $x$ with $(x - a)$ .
E.g. $y = x^{2}$ becomes $y = {(x - a)}^{2}$ .
Outline how would you find a new equation for a graph which has been translated vertically.
To find a new equation for a graph which has been translated vertically, consider how the function has been changed.
It has changed from $f (x)$ to $f (x) + a$ , this is a translation of $a$ units upwards.
Therefore in the equation of the graph, you can simply add on $a$
E.g. $y = x^{2}$ becomes $y = x^{2} + a$ .
Describe the transformation that has occurred when $y = x^{2}$ is transformed to ${(x - p)}^{2} + q$ .
When $y = x^{2}$ is transformed to ${(x - p)}^{2} + q$ , it has been translated $p$ units right, and $q$ units upwards.
This is why when completing the square, the vertex is at $(p, q)$ .
The graph of $y = f (x)$ is transformed to $y = - f (x)$ .
Describe the transformation that has occurred.
When the graph of $y = f (x)$ is transformed to $y = - f (x)$ , the graph has been reflected in the x-axis.
The graph of $y = f (x)$ is transformed to $y = f (- x)$ .
Describe the transformation that has occurred.
When the graph of $y = f (x)$ is transformed to $y = f (- x)$ , the graph has been reflected in the y-axis.
True or False?
Some graphs appear not to change when reflected.
True.
Some graphs appear not to change when reflected.
E.g. $y = x^{2}$ will look exactly the same when reflected in the y-axis.
Outline how would you find a new equation for a graph which has been reflected in the x-axis.
To find a new equation for a graph which has been reflected in the x-axis, consider how the function has been changed.
It has changed from $f (x)$ to $- f (x)$ .
Therefore you multiply the whole equation of the graph by -1.
E.g. $y = x^{2} - 7 x + 10$ becomes $y = - (x^{2} - 7 x + 10)$ or $y = - x^{2} + 7 x - 10$ .
Outline how would you find a new equation for a graph which has been reflected in the y-axis.
To find a new equation for a graph which has been reflected in the y-axis, consider how the function has been changed.
It has changed from $f (x)$ to $f (- x)$ .
Therefore you replace all the $x$ 's with $- x$ .
E.g. $y = x^{2} - 7 x + 10$ becomes $y = {(- x)}^{2} - 7 (- x) + 10$ or $y = x^{2} + 7 x + 10$ .
What is a stretch in the context of transformations of graphs?
In the context of transformations of graphs, a stretch is a transformation that enlarges or shrinks the graph in the $x$ -direction or $y$ -direction.
What transformation of the graph of $y = f (x)$ is indicated by the equation $y = f (a x) ?$
$y = f (a x)$ is a horizontal stretch (stretch in the $x$ -direction) by a scale factor of $\frac{1}{a}$ .
What transformation of the graph of $y = f (x)$ is indicated by the equation $y = a f (x) ?$
$y = a f (x)$ is a vertical stretch (stretch in the $y$ -direction) by a scale factor of $a$ .
True or False?
The equation $y = f (2 x)$ represents a horizontal stretch by a factor of $2$ .
False.
$y = f (2 x)$ represents a horizontal stretch by a factor of $\frac{1}{2}$ .
Remember that the scale factor of a stretch in the $x$ -direction is the reciprocal of the coefficient of $x$ .
What are invariant points?
Invariant points are points on a graph that do not change during a particular transformation.
What is the equation for a horizontal stretch of $y = f (x)$ by a factor of $\frac{1}{3}$ ?
The equation for a horizontal stretch of $y = f (x)$ by a factor of $\frac{1}{3}$ is $y = f (3 x)$ .
True or False?
$y = 3 f (x)$ represents a vertical stretch by a factor of 3.
True.
$y = 3 f (x)$ represents a vertical stretch by a factor of 3.
What is the equation for a vertical stretch of $y = f (x)$ by a factor of $4$ ?
The equation for a vertical stretch of $y = f (x)$ by a factor of $4$ is $y = 4 f (x)$ .
True or False?
Points on the $y$ -axis are invariant during a vertical stretch.
False.
Points on the $x$ -axis are invariant during a vertical stretch.

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Transformations of Graphs (Edexcel IGCSE Maths A)

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