IGCSEMathsCambridge (CIE)International MathsExtendedFlashcardsCoordinate Geometry & GraphsTransformations of Graphs

Transformations of Graphs (Cambridge (CIE) IGCSE International Maths: Extended): Flashcards

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FrontTranslations of Graphs
Define a translation of a graph.

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Cards in this collection (8)

Define a translation of a graph.
A translation is a transformation that slides a graph up, down, left or right in the $x y$ plane.
The shape, size and orientation of the graph are all unchanged, and only its position moves.
What does $y = f (x) + a$ do to the graph of $y = f (x)$ ?
It translates the whole graph vertically, without changing its shape at all.
The graph moves up when $a$ is positive and down when $a$ is negative.
How is a translation of a graph described using a vector?
As a column vector $[\begin{matrix} horizontal \\ vertical \end{matrix}]$ , giving the movement across and then the movement up.
So $[\begin{matrix} 0 \\ 1 \end{matrix}]$ is 1 unit up with no sideways movement, and $[\begin{matrix} 2 \\ 0 \end{matrix}]$ is 2 units across with no vertical movement.
True or False?
The graph of $y = f (x + 3)$ is the graph of $y = f (x)$ moved 3 units to the right.
False.
It moves 3 units to the left, which is the opposite of what most people expect.
A positive number added inside the bracket shifts the graph in the negative $x$ direction.
A translation always leaves one set of coordinates untouched. Complete both rules:
A vertical translation leaves every $______$ coordinate unchanged, while a horizontal translation leaves every $______$ coordinate unchanged instead.
The completed rules are:
A vertical translation leaves every $x$ coordinate unchanged, while a horizontal translation leaves every $y$ coordinate unchanged instead.
A point only moves in the direction of the translation, so its other coordinate has no reason to change.
The graph $y = x^{2} - 3 x + 7$ is transformed to $y = f (x - 6)$ . Write out the new equation.
Replace every $x$ in the equation by $(x - 6)$ , brackets included.
That gives $y = {(x - 6)}^{2} - 3 (x - 6) + 7$ , which may be left in that form or expanded to $y = x^{2} - 15 x + 61$ .
Which part of a translation does each letter carry in $y = f (x - p) + q$ ?
$p$ sits inside the bracket and carries the horizontal part, while $q$ sits outside it and carries the vertical part.
So $y = 3 x^{2}$ becoming $y = 3 {(x + 1)}^{2} + 2$ is one combined translation, horizontal and vertical at the same time.
What happens to a graph's asymptotes when the graph is translated?
They move with it, except that an asymptote parallel to the direction of travel is left where it is.
Moving $y = 2^{x}$ up by 3 lifts its horizontal asymptote from $y = 0$ to $y = 3$ , because that asymptote lies at right angles to the movement.

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