Question

answer the questions below to determine what kind of function is depicted in the table below.

x	0	1	2	3	4

answer attempt 1 out of 2

this function is because

Explanation:

Step1: Calculate x differences

$\Delta x = 1-0=2-1=3-2=4-3=1$

Step2: Calculate f(x) differences

$\Delta f(x) = -16-(-10)=-22-(-16)=-28-(-22)=-34-(-28)=-6$

Step3: Analyze the pattern

A constant first difference of $\Delta f(x)$ means the function has a constant rate of change, which is characteristic of a linear function. The general form of a linear function is $f(x)=mx+b$, where $m$ is the slope (equal to the constant difference) and $b$ is the y-intercept (value of $f(x)$ when $x=0$). Here, $m=-6$ and $b=-10$, so $f(x)=-6x-10$.

Answer:

This function is linear because the first differences of the $f(x)$ values are constant (equal to -6), indicating a constant rate of change.