Preview Activity 3.2.1.
Let \(a\text{,}\) \(h\text{,}\) and \(k\) be arbitrary real numbers with \(a \ne 0\text{,}\) and let \(f\) be the function given by the rule \(f(x) = a(x-h)^2 + k\text{.}\)
(a)
What familiar type of function is \(f\text{?}\) What information do you know about \(f\) just by looking at its form? (Think about the roles of \(a\text{,}\) \(h\text{,}\) and \(k\text{.}\))
(b)
Next we use some calculus to develop familiar ideas from a different perspective. To start, treat \(a\text{,}\) \(h\text{,}\) and \(k\) as constants and compute \(f'(x)\text{.}\)
(c)
Find all critical numbers of \(f\text{.}\) (These will depend on at least one of \(a\text{,}\) \(h\text{,}\) and \(k\text{.}\))
(d)
(e)
Based on the information you’ve found above, classify the critical values of \(f\) as maxima or minima.
