Question

Given that f(x)=4x² - 4x + 2,

a. Determine an expression for f(x + h) in terms of x and h.
b. Find the average rate of change of f over the interval [x, x+h]. Simplify as your answer.

Answer 1

The expression for f(x + h) is 4(x + h)² - 4(x + h) + 2. The average rate of change of f over the interval [x, x+h] is a linear function expressed as 8x + 4h - 4.

Step-by-step explanation:

To find the expression for f(x + h) in terms of x and h, we substitute x + h for x in the given function f(x)=4x² - 4x + 2. This yields:

• f(x + h) = 4(x + h)² - 4(x + h) + 2

Expanding this, we get:

• f(x + h) = 4(x² + 2xh + h²) - 4x - 4h + 2
• f(x + h) = 4x² + 8xh + 4h² - 4x - 4h + 2

To find the average rate of change of f over the interval [x, x+h], we use the formula:

• Average Rate of Change = (f(x + h) - f(x)) / h

Simplifying this expression:

• Average Rate of Change = ((4x² + 8xh + 4h² - 4x - 4h + 2) - (4x² - 4x + 2)) / h
• Average Rate of Change = (8xh + 4h² - 4h) / h
• Average Rate of Change = 8x + 4h - 4

So, the average rate of change of f over the interval is 8x + 4h - 4, which is a linear function of x and h.