Doug C. answered • 02/24/20
Math Tutor with Reputation to make difficult concepts understandable
Erin, do you feel pretty confident about squaring a binomial? For example can you expand (x + 5)2, quickly? There is a pattern which you should memorize. (a + b)2 = a2 + 2ab + b2. In words, square the first term, double the product of the two terms, then square the 2nd term. So, (x + 5)2 = x2 + 2(5)(x) + 52 = x2 + 10x + 25. The same applies if the two terms of the binomial are separated by a minus sign. So, (x - 7)2 = x2 - 14x + 49. You should practice several of these until it comes easily for you.
So, f(a) = -3 + 4(a)2
f(a+h) = -3 +4(a+h)2 = -3 +4(a2+2ah + h2) = -3 +4a2+8ah+4h2 (now you see why I included the first paragraph)
f(a+h) - f(a) = (-3 + 4a2+8ah+4h2) - (-3 + 4a2) = 8ah+4h2 (removed the grouping symbols and combined similar terms.
Finally, (f(a+h) - f(a)) / h = (8ah + 4h2) / h = (8a + 4h) (factor an h from each term of the numerator and cancel with the h in the denominator. (Or, think divide h into both terms of the numerator).
FYI, the expression (f(x+h) - f(x))/h is called a difference quotient and will be one of the first concepts you learn about when you start calculus.
