Question

find each of the following for f(x) = 3x² - 6x + 7. (a) f(x + h) (b) f(x + h) - f(x) (c) f(x + h) - f(x) / h (a) f(x + h) = □

Explanation:

Step1: Substitute \( x + h \) into \( f(x) \)

We have \( f(x) = 3x^2 - 6x + 7 \), so replace \( x \) with \( x + h \):
\( f(x + h) = 3(x + h)^2 - 6(x + h) + 7 \)

Step2: Expand \( (x + h)^2 \)

Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), we get \( (x + h)^2 = x^2 + 2xh + h^2 \). Substitute this into the expression:
\( f(x + h) = 3(x^2 + 2xh + h^2) - 6(x + h) + 7 \)

Step3: Distribute the constants

Distribute 3 and -6:
\( f(x + h) = 3x^2 + 6xh + 3h^2 - 6x - 6h + 7 \)

Answer:

\( 3x^2 + 6xh + 3h^2 - 6x - 6h + 7 \)