Grade 11

Functions & Graphs

Go deeper into functions — understand how transformations shift, stretch and flip graphs, and explore logs.

Function Transformations

Any function $f(x)$ can be transformed by wrapping it with three parameters — $a$ , $p$ and $q$ — to get a new function:

Theorem

g(x) = a \cdot f(x - p) + q

$a$ — vertical stretch/flip. $|a|>1$ makes it taller; $0<|a|<1$ squishes it; $a<0$ flips it upside down.
$p$ — horizontal shift. $p>0$ moves it right; $p<0$ moves it left.
$q$ — vertical shift. $q>0$ moves it up; $q<0$ moves it down.

Interactive — choose a function type and adjust a, p, q

Try it yourself

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The gray curve in the widget is the original untransformed function. Watch how each slider affects the graph.

Worked Example

Describe in words how $g(x) = -2(x+1)^2 - 3$ is related to $f(x) = x^2$