# What is the range of the function f(x) = 9x^2 - 9x?

Apr 23, 2018

$\left[- \frac{9}{4} , \infty\right)$

#### Explanation:

$\text{since the leading coefficient is positive }$

$f \left(x\right) \text{ will be a minimum } \bigcup$

$\text{we require to find the minimum value}$

$\text{find the zeros by setting } f \left(x\right) = 0$

$\Rightarrow 9 {x}^{2} - 9 x = 0$

$\text{take out a "color(blue)"common factor } 9 x$

$\Rightarrow 9 x \left(x - 1\right) = 0$

$\text{equate each factor to zero and solve for x}$

$9 x = 0 \Rightarrow x = 0$

$x - 1 = 0 \Rightarrow x = 1$

$\text{the axis of symmetry is at the midpoint of the zeros}$

$\Rightarrow x = \frac{0 + 1}{2} = \frac{1}{2}$

$\text{substitute this value into the equation for minimum value}$

$y = 9 {\left(\frac{1}{2}\right)}^{2} - 9 \left(\frac{1}{2}\right) = \frac{9}{4} - \frac{9}{2} = - \frac{9}{4} \leftarrow \textcolor{red}{\text{min. value}}$

$\Rightarrow \text{range } y \in \left[- \frac{9}{4} , \infty\right)$
graph{9x^2-9x [-10, 10, -5, 5]}