Quadratic Functions and Vertex Form

Definition A quadratic function is a function of the form

where $a$, $b$ and $c$ are real constants and $a \ne 0$. Its domain is all real numbers unless a smaller domain is specified, and its graph is a curve called a parabola. The parabola opens upward when $a > 0$ and downward when $a < 0$.

Definition The standard form of a quadratic function is $f(x) = ax^2 + bx + c$ with $a \ne 0$. The vertex form is

Every quadratic function can be written in both forms with the same $a$, as completing the square below shows, and the numbers $h$ and $k$ are

In vertex form the graph is the graph of $y = x^2$ scaled vertically by the factor $\lvert a \rvert$, reflected across the $x$-axis when $a < 0$, and then shifted $h$ units to the right and $k$ units up. Some books call the vertex form the standard form and call $ax^2 + bx + c$ the general form, so check which convention a source uses.

Definition The vertex of the parabola $y = a(x - h)^2 + k$ is the point $(h, k)$. Because $(x - h)^2 \ge 0$, with equality only at $x = h$, the number $k$ is the minimum value of $f$ over all real $x$ when $a > 0$ and the maximum value when $a < 0$, and $f$ reaches it only at $x = h$. On a smaller domain that leaves out $h$, the value $k$ is never reached.

Definition The axis of symmetry of the parabola $y = a(x - h)^2 + k$ is the vertical line $x = h$. The graph is symmetric about this line, which means

because both sides equal $ad^2 + k$.

Definition For every real number $b$,

To put $f(x) = ax^2 + bx + c$ in vertex form, factor $a$ out of the two terms with $x$, complete the square inside the parentheses with this identity, and then collect the constants.

Definition A real zero of a function $f$ is a real number $r$ with $f(r) = 0$. The real zeros are the first coordinates of the $x$-intercepts $(r, 0)$ of the graph. Factoring a quadratic means writing it as

with real numbers $r$ and $s$, which is possible exactly when the quadratic has at least one real zero. A product of real numbers is zero only when one of the factors is zero, so the real zeros are then exactly $r$ and $s$, a single number when $r = s$.

Definition Let $a$, $b$ and $c$ be real numbers with $a \ne 0$. If $b^2 - 4ac \ge 0$, the real solutions of $ax^2 + bx + c = 0$ are

and if $b^2 - 4ac < 0$ the equation has no real solutions. The formula comes from completing the square in general, which turns the equation into

Definition The discriminant of $ax^2 + bx + c$ with $a \ne 0$ is the number $D = b^2 - 4ac$. It sorts quadratics into three cases.

• When $D > 0$ there are two distinct real zeros, and the graph crosses the $x$-axis twice.
• When $D = 0$ there is exactly one real zero, $x = -\frac{b}{2a}$, and the vertex lies on the $x$-axis.
• When $D < 0$ there are no real zeros, and the graph never meets the $x$-axis. The equation then has two solutions that are not real numbers, which come up in the lesson on complex numbers.

Definition When a quadratic function models a situation, the model holds only on a domain set by that situation, such as nonnegative times or positive lengths. A vertex or zero computed from the formula answers the question only if it lies in that domain, and every answer should be checked against the original conditions and units.