Given the function $f\left(x\right)=2x^2-3x+1$, which of the following is the average rate of change of $f\left(x\right)$ over the interval from $x=-6$ to $x=-3$?

• A. $9$
• B. $21$
• C. $15$
• D. $27$

Answer: (D)

Verified step by step guidance

1. Step 1: Recall the formula for the average rate of change of a function f(x) over an interval [a, b]. It is given by: $\frac{\left(f_b-f_a\right)}{\left(b-a\right)}$, where f(b) and f(a) are the function values at x = b and x = a, respectively.
2. Step 2: Identify the interval [a, b] from the problem. Here, a = -6 and b = -3. Substitute these values into the formula.
3. Step 3: Compute f(a) and f(b) using the given function f(x) = 2x^2 - 3x + 1. For f(-6), substitute x = -6 into the function: $f_{-6}=2(-6{)}^2-3(-6)+1$. Similarly, compute f(-3): $f_{-3}=2(-3{)}^2-3(-3)+1$.
4. Step 4: Substitute the computed values of f(-6) and f(-3) into the formula for the average rate of change: $\frac{\left(f_{-3}-f_{-6}\right)}{\left(-3--6\right)}$.
5. Step 5: Simplify the numerator and denominator to find the average rate of change. The numerator is the difference between f(-3) and f(-6), and the denominator is the length of the interval, which is $-3--6$. Perform the arithmetic operations to complete the calculation.