Advanced Math / Nonlinear Functions
Nonlinear Functions
Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, domain restrictions, or the meaning of parameters in context.
Least you need to know
- For $f(x)=a(x-h)^2+k$, the vertex is $(h,k)$.
- A quadratic with positive leading coefficient has a minimum; one with negative leading coefficient has a maximum.
- For an exponential function $f(x)=ab^x$, $a$ is the initial value and $b$ is the growth or decay factor.
Key notation
- $f(x)$ — The output of function $f$ for input $x$
- $a(x-h)^2+k$ — Vertex form of a quadratic function
Worked example
- Consider $f(x)=2(x-3)^2+5$.
- The expression $(x-3)^2$ is always at least 0.
- Therefore, $2(x-3)^2+5$ is at least 5.
- The minimum value of $f$ is 5, occurring when $x=3$.
Common mistakes
- In vertex form, $x-h$ means the vertex has $x$-coordinate $h$, not $-h$.
- A function value such as $f(4)$ asks for an output, not a product.
- For exponential decay, the growth factor is between 0 and 1.
How to recognize it
- The function includes powers, roots, exponentials, or a variable in a denominator.
- The problem may ask for a value such as $f(3)$, a vertex, an intercept, a maximum or minimum, or the meaning of a parameter.
- The function may be written in a form where its key feature is visible without expanding.
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