Question

1. here is the graph that represents an absolute value function. which equation below represents the function?

a. $f(x) = |x| - 1$
b. $f(x) = |x - 1|$
c. $f(x) = |x + 1|$
d. $f(x) = |x| + 1$

1. the table and the graph show the population of a country between 2010 and 2015.

year	population (millions)
2011	37.7
2012	38.2
2013	38.4
2014	38.9
2015	39.1

a. find the average rate of change in population from 2010 to 2015.
b. what is the domain of this function?
c. what is the range of this function?

Explanation:

Question 5

Step1: Recall absolute value function vertex

The vertex of \( f(x) = |x - h| + k \) is \((h, k)\). The graph's vertex is at \((1, 0)\).

Step2: Analyze each option

• Option a: \( f(x) = |x| - 1 \) has vertex \((0, -1)\) – incorrect.
• Option b: \( f(x) = |x - 1| \) has vertex \((1, 0)\) – matches the graph.
• Option c: \( f(x) = |x + 1| = |x - (-1)| \) has vertex \((-1, 0)\) – incorrect.
• Option d: \( f(x) = |x| + 1 \) has vertex \((0, 1)\) – incorrect.

Step1: Identify formula for average rate of change

The formula for average rate of change is \( \frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \). Here, \( x_1 = 2010 \), \( f(x_1) = 36.5 \); \( x_2 = 2015 \), \( f(x_2) = 39.1 \).

Step2: Calculate the change

\( \Delta y = 39.1 - 36.5 = 2.6 \), \( \Delta x = 2015 - 2010 = 5 \). Then, average rate of change \( = \frac{2.6}{5} = 0.52 \) (in millions per year).

The domain of a function representing population over time (years 2010–2015) includes all input values (years). So the domain is the set of years from 2010 to 2015, or in interval notation, \( [2010, 2015] \).

Answer:

b. \( f(x) = |x - 1| \)

Question 6a