Hi Jamey,

The variable in the parentheses tells you what you are substituting for x in the function.

In the first case, f(a), you substitute x = a.

So, if f(x) = 7 -x + 14x^{2}

Then f(a) = 7 -a + 14a^{2} or f(a) = 14a^{2} - a + 7

**(The convention is to write algebraic expressions in decreasing order of powers)**

For f(a+h) we substitute x = a+h.

f(x) = 14x^{2 }- x + 7

f(a+h) = 14 (a+h)^{2} - (a+h) + 7

= **14(a**^{2} + 2ah + h^{2}) - a - h + 7

**Note: Make sure to multiply ALL terms in the parentheses by 14!**

f(a+h) = 14a^{2 }+ 28ah + 14h^{2} - a - h + 7

Next we can calculate f(a+h) - f(a),

f(a+h) - f(a) = 14a^{2} + 28ah + 14h^{2} - a - h + 7 **
- (14a**^{2} - a + 7)

**Note: Make sure to multiply ALL terms in the parentheses by -1**

= 14a^{2} + 28ah + 14h^{2} - a - h + 7 - 14a^{2} + a - 7

= 14a^{2} + 28ah + 14h^{2}
- a - h + 7 - 14a^{2} + a
- 7

The 14a^{2} and a terms cancel out and so do the 7's to leave us with:

f(a+h) - f(a) = 28ah + 14h^{2} - h = 14h^{2} + 28ah - h

(f(a+h) - f(a)) /h = 14h - 28a - 1 = 14(h - 2a) - 1

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