Advanced Math / Nonlinear Functions

Nonlinear Function Transformations and Features

Nonlinear function questions may involve transformed functions, vertices, zeros, symmetry, evaluating composite-looking expressions, or comparing function features from different forms.

Least you need to know

For $g(x)=f(x-h)+k$, the graph of $f$ is shifted right $h$ units and up $k$ units.
For $a(x-h)^2+k$, the vertex is $(h,k)$.
If $f(x)=(x-r)(x-s)$, the zeros are $r$ and $s$.
To evaluate $f(g(a))$, first find $g(a)$, then use that result as the input to $f$.

Key notation

$g(x)=f(x-3)+2$ — A transformation of $f$ shifted right 3 and up 2
$f(g(2))$ — Evaluate $g(2)$ first, then substitute into $f$

Worked example

If $f(x)=x^2$ and $g(x)=f(x-4)+1$, then $g(x)=(x-4)^2+1$.
The vertex of $g$ is $(4,1)$.

Common mistakes

The expression $x-h$ corresponds to a shift right by $h$.
The output shift changes the $y$-value, not the input.
For composition-style evaluation, preserve the order of operations.

How to recognize it

The problem defines a function in terms of another function.
The problem asks for a vertex, zero, maximum, minimum, or transformed value.
The problem uses notation such as $f(g(a))$ or $g(x)=f(x-h)+k$.

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Nonlinear Functions

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