Describing Quadratic Functions

We want to identify the vertex, axis of symmetry, absolute maximum or minimum value, domain, and range of the given quadratic function. To do so, we will first express it in vertex form, $y=a(x-h{)}^2+k,$ where $a,$ $h,$ and $k$ are either positive or negative constants. $$\begin{array}{c}y=\text{-}(x-4{)}^2-25\\ \Updownarrow \\ y=\text{-}1(x-4{)}^2+(\text{-}25)\end{array}$$ It is important to note that we do not need to graph the parabola to identify the desired information. Let's compare the general formula for the vertex form to our equation. $$\begin{array}{rl}\text{General formula: }y=& a(x-h{)}^2+k\\ \text{Equation: }y=& \text{-}1(x-4{)}^2+(\text{-}25)\end{array}$$ We can see that $a=\text{-}1,$ $h=4,$ and $k=\text{-}25.$

Vertex

The vertex of a quadratic function written in vertex form is the point $(h,k).$ For this exercise, we have $h=4$ and $k=\text{-}25.$ Therefore, the vertex of the given equation is $(4,\text{-}25).$

Axis of symmetry

The axis of symmetry of a quadratic function written in vertex form is the vertical line with equation $x=h.$ As we have already noticed, for our function, this is $h=4.$ Thus, the axis of symmetry is the line $x=4.$

Maximum or minimum value

Before we determine the maximum or minimum recall that, if $a>0,$ the parabola opens upwards. Conversely, if $a<0,$ the parabola opens downwards.

In the given function, we have $a=\text{-}1,$ which is less than $0.$ Thus, the parabola opens downwards and we will have a maximum value. The minimum or maximum value of a parabola is always the $y\text{-}$coordinate of the vertex, $k.$ For this function, it is $k=\text{-}25.$

Domain and Range

Unless there is a specific restriction given in the context of the problem, the domain of a quadratic function is all real numbers. In this case, there is no restriction on the value of $x.$ Since the maximum value of the function is $\text{-}25,$ the range is all real numbers less than or equal to $\text{-}25.$ $$\begin{array}{cc}\text{Range:}& y\le \text{-}25\end{array}$$