F(x)=3x^2 -x +2
Find F(a+h)
To find F(a+h), substitute (a+h) into the expression for F(x):
F(a+h) = 3(a+h)^2 - (a+h) + 2
Expanding the expression and combining like terms:
F(a+h) = 3(a^2 + 2ah + h^2) - a - h + 2
F(a+h) = 3a^2 + 6ah + 3h^2 - a - h + 2
So, F(a+h) = 3a^2 + 6ah + 2h^2 - (a + h) + 2.
How did 3h^2 go to 2h^2?
Apologies for the mistake. You are correct, the coefficient of h^2 should still be 3. So the correct expression is:
F(a+h) = 3a^2 + 6ah + 3h^2 - (a + h) + 2
To find F(a+h), you need to substitute (a+h) for x in the given function.
F(x) = 3x^2 - x + 2
Replacing x with (a+h):
F(a+h) = 3(a+h)^2 - (a+h) + 2
Now, let's simplify this expression step-by-step:
1. Expand the square: (a+h)^2 = (a+h)(a+h) = a^2 + 2ah + h^2
F(a+h) = 3(a^2 + 2ah + h^2) - (a+h) + 2
2. Distribute 3 to each term inside the parentheses:
F(a+h) = 3a^2 + 6ah + 3h^2 - (a+h) + 2
3. Distribute -1 to each term inside the parentheses:
F(a+h) = 3a^2 + 6ah + 3h^2 - a - h + 2
4. Combine like terms:
F(a+h) = 3a^2 + 6ah - a + 3h^2 - h + 2
So, F(a+h) = 3a^2 + 6ah - a + 3h^2 - h + 2 is the simplified expression for F(a+h).
To find F(a+h), we need to substitute (a+h) in place of x in the given function F(x) = 3x^2 - x + 2.
So, F(a+h) = 3(a+h)^2 - (a+h) + 2.
To simplify this, we need to expand the expression (a+h)^2.
(a+h)^2 = (a+h)(a+h)
= a(a+h) + h(a+h)
= a^2 + ah + ah + h^2
= a^2 + 2ah + h^2.
Now, substitute this expanded expression back into F(a+h):
F(a+h) = 3(a^2 + 2ah + h^2) - (a+h) + 2
= 3a^2 + 6ah + 3h^2 - a - h + 2.
This is the simplified expression for F(a+h).