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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