Advanced Math / Nonlinear Functions

Polynomial Function Zeros and Features

Polynomial function questions often use factored form to identify zeros and standard form to identify intercepts or evaluate function values. Parameter questions may ask for a coefficient or constant from a known zero or value.

Least you need to know

Zeros are the input values that make the function equal to zero.
The $y$-intercept is the value of the function when $x=0$.
If $x=r$ is a zero, then $(x-r)$ is a factor of the polynomial.
Function notation such as $f(2)$ means substitute 2 for the input.

Key notation

$f(x)=(x-4)(x+1)(x-2)$ — A polynomial function with zeros $4$, $-1$, and $2$
$p(0)$ — The $y$-intercept of the graph of $y=p(x)$

Worked example

If $f(x)=(x-3)(x+5)$, then $f(3)=0$ and $f(-5)=0$.
Therefore, the zeros of $f$ are $3$ and $-5$.

Common mistakes

A factor $x+4$ gives a zero of $-4$.
The $y$-intercept is not necessarily a zero.
For parameter questions, substitute the known input and output before solving.

How to recognize it

The problem asks for a zero, intercept, or function value.
The function is written in factored form.
The problem gives a known zero and asks for a parameter.

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Quadratic Function Features