Quick questions on Transformations, symmetry and composite functions for HSC Maths Advanced: translating and reflecting a known curve, even and odd functions with the algebraic test $f(-x) = f(x)$ and $f(-x) = -f(x)$ , sketching absolute-value graphs from $y = |x|$ , and building composite functions $f(g(x))$ and finding their domain, with worked examples and practice questions

2short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is translating a known graph?

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A translation slides a graph without rotating, reflecting or resizing it: every point moves the same distance in the same direction. There are two independent moves, vertical and horizontal, and the rules look pleasingly similar once you see where each one acts.

What is absolute-value graphs by transformation?

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The absolute value

|x|

is the size of

x

ignoring its sign, that is, its distance from

0

on the number line:

|3| = 3

and

|-3| = 3

. Written in cases,

|x| = x

for

x \ge 0

and

|x| = -x

for

x < 0

, two straight lines of gradient

+1

and

-1

that meet at the origin. So the graph of

y = |x|

is a V-shape with its sharp corner (vertex) at

(0, 0)

, sitting on or above the

x

-axis; it is even, with the

y

-axis as its mirror line.

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