Answer
(a) See below.
(b) Domain: $(-\infty, \infty)$
Range:$(-\infty, 0]$
(c) Domain where function increases:$(-\infty, \frac{5}{2})$
Domain where function decreases: $(\frac{5}{2}, \infty)$
Work Step by Step
First, we convert to vertex form to find the range and increasing and decreasing domains easier:
$f(x)=-4x^2+20x-25$
$f(x)=-4(x^2-5x)-25$
$f(x)=-4(x^2-5x+\frac{25}{4})-25+25$
$f(x)=-4(x-\frac{5}{2})^2$
Recap:
The domain is a horizontal span from the function's smallest value of x to the function's largest value of x.
The range is a vertical span from the function's smallest value of f(x) to the function's largest value of f(x).
A function is increasing in the domain intervals where its slope is positive. On the other hand, a function is decreasing in the domain intervals where its slope is negative. In a quadratic function, we can find where the function starts or stops increasing by locating the vertex.