Question

an object traveling with an initial constant speed along the y - axis begins to decelerate 4 kilometers before reaching a reference position. its position is given by the function ( y = t^2 - 2t - 4 ), where ( t ) is the time in seconds. which value is equal to the average rate of change of the function over the interval ( (0, - 4) ) to ( (3, - 1) )?
a. ( - 4 )
b. ( - 1 )
c. ( 1 )
d. ( 3 )

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \( y = f(t) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = 1 \), \( b = 3 \), and \( f(t)=t^{2}-3t - 4 \).

Step2: Calculate \( f(1) \)

Substitute \( t = 1 \) into \( f(t) \):
\( f(1)=(1)^{2}-3(1)-4=1 - 3 - 4=-6 \)

Step3: Calculate \( f(3) \)

Substitute \( t = 3 \) into \( f(t) \):
\( f(3)=(3)^{2}-3(3)-4=9 - 9 - 4=-4 \)

Step4: Apply the average rate of change formula

Using the formula \(\frac{f(3)-f(1)}{3 - 1}\), substitute the values of \( f(3) \) and \( f(1) \):
\(\frac{-4-(-6)}{3 - 1}=\frac{-4 + 6}{2}=\frac{2}{2}=1\)

Answer:

C. 1