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Question

Find f(a),

 f(a + h),

and the difference quotient f(a + h) − f(a)

h

,

 where h ≠ 0.

f(x) = 6x² + 8

Answer 1

f(a+h) = 6(a+h)^2 + 8 =

6(a^2+2ah+h^2) + 8 =

6a^2 + 12ah + 6h^2 + 8

subtracting f(a) gives the

numerator of the difference

quotient, f(a+h)-f(a)

It is 12ah + 6h^2

dividing that by h : 12a + 6h

Answer 2

1) We are given a function f(x) = 6x² + 8. In order to find f(a), I am going to plug in a everywhere I see an x. Thus, since there is an x in 6x², I am going to plug a in there. Therefore, f(a) = 6a² + 8.

2) Next, we are going to do the same for f(a + h). This gives us f(a+h) = 6(a+h)² + 8. In order to get the answer though, we will need to expand (a+h)². Expanding will give us a² + 2ah + h². Finally, we distribute the 6 into this polynomial. So f(a+h) = 6a² + 12ah + 6h² + 8

3) Finally, we will find the difference quotient f(a+h) - f(a). Plugging in our answers from parts 1 and 2, we get f(a+h) - f(a) = (6a² + 12ah + 6h² + 8) - (6a² + 8). Using the distributive property, this becomes 6a² + 12ah + 6h² + 8 - 6a² - 8, which will simplify to 12ah + 6h². Hence f(a+h)- f(a) = 12ah + 6h².