Question

Compare the equations represented in the table, equation, and graph over
the interval
[-5, 3]. Which function is increasing the fastest?

Answer 1

Answer:

Tabled Function

Step-by-step explanation:

To determine which function is increasing the fastest over the interval [-5, 3], we need to calculate and compare each function's average rate of change over the given interval.

The average rate of change of function f(x) over the interval a ≤ x ≤ b is given by:

$\dfrac{f(b)-f(a)}{b-a}$

Given interval:  -5 ≤ x ≤ 3

Therefore, a = -5  and  b = 3

Tabled function

$f(3)=7$

$f(-5)=-17$

$\implies \textsf{Average rate of change}=\dfrac{f(3)-f(-5)}{3-(-5)}=\dfrac{7-(-17)}{3+5}=3$

Equation:  y = x² - 2

$f(3)=(3)^2-2=7$

$f(-5)=(-5)^2-2=23$

$\implies \textsf{Average rate of change}=\dfrac{f(3)-f(-5)}{3-(-5)}=\dfrac{7-23}{3-(-5)}=-2$

Graphed function

From inspection of the graph:

$f(3)\approx8$

$f(-5) \approx 0$

$\implies \textsf{Average rate of change}=\dfrac{f(3)-f(-5)}{3-(-5)} \approx \dfrac{8-0}{3-(-5)}=1$

Therefore, the Tabled Function has the greatest average rate of change in the interval [-5, 3] and so it is increasing the fastest.