# What is the domain and range of -(3/25)(x-26)^2?

Aug 16, 2015

Domain: $\left(- \infty , + \infty\right)$
Range: $\left(0 , - \infty\right)$

#### Explanation:

Your function is defined for any value of $x \in \mathbb{R}$, which means that its domain will be $\left(- \infty , + \infty\right)$.

In order to find the function's range, you need to take into account the fact that the aquare of an expression will always be positive, regardless of the sign of said expression.

This means that ${\left(x - 26\right)}^{2} > 0 \text{, } \left(\forall\right) x \in \mathbb{R}$. An important consequence is that this expression is minimum when $x = 26$.

Since you have a negative number multiplied by a square, the function will always have negative values, the only exception being the case when $x - 26 = 0$, for which

$y = - \frac{3}{25} \cdot {0}^{2} = 0$

This means that the range of the function will essentially be $\left(0 , - \infty\right)$.

graph{-3/25 * (x-26)^2 [-40.84, 90.8, -48.74, 17.1]}