Question

question
find the average rate of change of the function
f(x), given below, from x = 1 to x = t.
$$ f ( x ) = - 2 x ^ { 2 } - 3 x - 2 $$
give your answer in terms of t.
provide your answer below:

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) from \(x=a\) to \(x = b\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 1\) and \(b=t\).

Step2: Calculate \(f(1)\) and \(f(t)\)

• For \(f(1)\):

Substitute \(x = 1\) into \(f(x)=-2x^{2}-3x - 2\).
\(f(1)=-2(1)^{2}-3(1)-2=-2 - 3-2=-7\).

• For \(f(t)\):

Substitute \(x = t\) into \(f(x)=-2x^{2}-3x - 2\).
\(f(t)=-2t^{2}-3t - 2\).

Step3: Substitute into the average - rate - of - change formula

\(\frac{f(t)-f(1)}{t - 1}=\frac{(-2t^{2}-3t - 2)-(-7)}{t - 1}\).
Simplify the numerator: \((-2t^{2}-3t - 2 + 7)=-2t^{2}-3t + 5\).
Factor the numerator: \(-2t^{2}-3t + 5=-(2t^{2}+3t - 5)=-(2t + 5)(t - 1)\).
So, \(\frac{-(2t + 5)(t - 1)}{t - 1}\), and since \(t
eq1\) (denominator cannot be zero), we can cancel out the \((t - 1)\) terms.

Answer:

\(-2t-5\)