True or False? Translating a graph changes its position but not its shape, its size or which way up it is.

True. A translation slides every point of the graph by the same amount in the same direction, so the curve keeps exactly the same shape, size and orientation.

GCSEMathsOCRHigherFlashcards7. Graphs of Equations & FunctionsTransformations of Graphs

Transformations of Graphs (OCR GCSE Maths: Higher): Flashcards

Exam code: J560

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FrontTranslations of Graphs
How is the graph of $y = x^{2} + 2$ related to the graph of $y = x^{2}$ ?

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How is the graph of $y = x^{2} + 2$ related to the graph of $y = x^{2}$ ?
The graph of $y = x^{2} + 2$ is the graph of $y = x^{2}$ translated 2 units up, by the vector $(\begin{matrix} 0 \\ 2 \end{matrix})$ .
Adding 2 to the whole right-hand side adds 2 to every $y$ -coordinate, while every $x$ -coordinate stays the same.
True or False?
The graph of $y = (x + 3)^{2}$ is the graph of $y = x^{2}$ translated 3 units to the right.
False.
The graph moves 3 units to the left, by the vector $(\begin{matrix} - 3 \\ 0 \end{matrix})$ .
Adding 3 inside the bracket means $x$ must be 3 smaller to give the same output as before, so every point moves 3 units left.
Fill in the gaps to give the new equation when $y = x^{2} - 3 x + 7$ is translated 6 units to the right:
$y = {(______)}^{2} - 3 (______) + 7$
The completed equation is:
$y = (x - 6)^{2} - 3 (x - 6) + 7$
Every $x$ in the equation is replaced by $(x - 6)$ , and the result can be expanded to $y = x^{2} - 15 x + 61$ if needed.
Describe the single transformation that maps $y = x^{3}$ onto $y = (x - 4)^{3} - 6$ .
The transformation is a translation by the vector $(\begin{matrix} 4 \\ - 6 \end{matrix})$ , which moves the graph 4 units right and 6 units down.
The $- 4$ inside the bracket moves the graph right and the $- 6$ outside the bracket moves it down.
True or False?
Translating a graph changes its position but not its shape, its size or which way up it is.
True.
A translation slides every point of the graph by the same amount in the same direction, so the curve keeps exactly the same shape, size and orientation.
What is the equation of the graph of $y = 4 x^{2} + 2 x + 1$ after a translation of 5 units down?
Subtract 5 from the whole right-hand side, which gives $y = 4 x^{2} + 2 x + 1 - 5$ at first.
So the new equation is $y = 4 x^{2} + 2 x - 4$ .
What is the equation of the graph of $y = x^{2} + 2 x$ after a reflection in the $x$ -axis?
The new equation is $y = - (x^{2} + 2 x)$ , which simplifies to $y = - x^{2} - 2 x$ .
Every $y$ -coordinate changes sign, so the whole right-hand side is multiplied by $- 1$ .
True or False?
The graph of $y = - x^{2}$ is the reflection of the graph of $y = x^{2}$ in the $y$ -axis.
False.
The graph of $y = - x^{2}$ is the reflection in the $x$ -axis, because the minus sign outside changes the sign of every $y$ -coordinate.
A reflection in the $y$ -axis would replace $x$ with $(- x)$ instead.
Fill in the missing letters:
Reflecting a graph in the $y$ -axis changes the sign of every $______$ coordinate and leaves every $______$ coordinate unchanged.
The completed sentence is:
Reflecting a graph in the $y$ -axis changes the sign of every $x$ coordinate and leaves every $y$ coordinate unchanged.
So a point such as $(2, - 3)$ moves to $(- 2, - 3)$ , and points on the $y$ -axis do not move at all.
What is the equation of the graph of $y = x^{2} + 2 x$ after a reflection in the $y$ -axis?
Replace every $x$ with $(- x)$ , which gives $y = (- x)^{2} + 2 (- x)$ at first.
This simplifies to $y = x^{2} - 2 x$ .
True or False?
Reflecting the graph of $y = \cos x$ in the $y$ -axis gives exactly the same graph.
True.
The graph of $y = \cos x$ is symmetrical about the $y$ -axis, so reflecting it in that axis maps it onto itself.
This is why $y = \cos (- x)$ and $y = \cos x$ have the same graph.
The graph of $y = x^{3}$ is reflected in the $x$ -axis and then translated 3 units to the left. What is the equation of the new graph?
Reflecting in the $x$ -axis gives $y = - x^{3}$ , and translating 3 units left then replaces $x$ with $(x + 3)$ .
So the new equation is $y = - (x + 3)^{3}$ .

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