Any quadratic function has either a maximum or a minimum value. We can tell whether a quadratic function has a maximum or a minimum by observing the vertex of its graph.

Here are graphs representing functions $f$ and $g$, defined by $f(x)=\text-(x+5)^2+4$ and $g(x)=x^2+6x-1$.

We know that a quadratic expression in vertex form can reveal the vertex of the graph, so we don’t actually have to graph the expression. But how do we know, without graphing, if the vertex corresponds to a maximum or a minimum value of a function?

The vertex form can give us that information as well!

To see if $(\text-3, \text-10)$ is a minimum or maximum of $g$, we can rewrite $x^2+6x-1$ in vertex form, which is $(x+3)^2-10$. Let’s look at the squared term in $(x+3)^2-10$.

• When $x=\text-3$, $(x+3)$ is 0, so $(x+3)^2$ is also 0.
• When $x$ is not -3, the expression $(x+3)$ is a nonzero number, and $(x+3)^2$ is positive.
• Because a squared number cannot have a value less than 0, $(x+3)^2$ has the least value when $x=\text{-}3$.

To see if $(\text-5,4)$ is a minimum or maximum of $f$, let’s look at the squared term in $\text-(x+5)^2+4$.

• When $x=\text-5$, $(x+5)$ is 0, so $(x+5)^2$ is also 0.
• When $x$ is not -5, the expression $(x+5)$ is nonzero, so $(x+5)^2$ is positive. The expression $\text-(x+5)^2$ has a coefficient of -1, however. Multiplying $(x+5)^2$ (which is positive when $x \neq \text-5$) by a negative number results in a negative number.
• Because a negative number is always less than 0, the value of $\text-(x+5)^2+4$ will always be less when $x \neq \text-5$ than when $x=\text-5$. This means $x =\text-5$ gives the greatest value of $f$.