Advanced Math / Nonlinear Functions

Quadratic Function Features

Quadratic functions can be interpreted through vertex, standard, and factored forms. Vertex form gives the vertex and maximum or minimum value, while factored form gives the zeros and the axis of symmetry.

Least you need to know

In $a(x-h)^2+k$, the vertex is $(h,k)$.
If $a>0$, the vertex gives a minimum value; if $a<0$, it gives a maximum value.
In $a(x-r)(x-s)$, the zeros are $r$ and $s$.
The axis of symmetry is halfway between the two zeros or is $x=-\frac{b}{2a}$ for $ax^2+bx+c$.

Key notation

$f(x)=(x-5)^2-7$ — A quadratic function with vertex $(5,-7)$
$h(x)=(x-2)(x+4)$ — A quadratic function with zeros 2 and -4

Worked example

For $f(x)=-(x+4)^2+11$, the vertex is $(-4,11)$.
Since the coefficient of the squared term is negative, the parabola opens downward.
Therefore, the maximum value of $f$ is $11$.

Common mistakes

In vertex form, $x-h$ corresponds to an $x$-coordinate of $h$, not $-h$.
The zeros come from setting each factor equal to zero.
The axis of symmetry is a line, so it is written as $x=$ a number.

How to recognize it

The problem asks for the vertex, maximum, minimum, zeros, or axis of symmetry.
The function is given in vertex or factored form.
The problem gives roots or function features and asks for a parameter.

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Radical and Rational Function Domains