Question

this graph shows the number of active phone land lines in millions since 2002.
phone land lines
(0, 50.5)
50
(1, 35.35)
40
(2, 24.745)
30
20
10
0 1 2 3 4 5 6 7 8 9 10
number of active
land lines (millions)
number of years since 2002
based on this information, which function best models the number of active phone land lines in millions x years since 2002?
a ( h(x) = 10.605(50.5)^x )

Explanation:

Step1: Identify the type of function

The graph shows a decreasing curve, so it's likely an exponential decay function. The general form of an exponential decay function is \( h(x) = a(b)^x \), where \( 0 < b < 1 \), and \( a \) is the initial value (when \( x = 0 \)).

Step2: Find the initial value \( a \)

When \( x = 0 \), the point is \( (0, 50.5) \). So, substituting \( x = 0 \) into \( h(x) = a(b)^x \), we get \( h(0) = a(b)^0 = a(1) = a \). Thus, \( a = 50.5 \).

Step3: Find the base \( b \)

Use the point \( (1, 35.35) \). Substitute \( x = 1 \), \( h(1) = 35.35 \), and \( a = 50.5 \) into \( h(x) = 50.5(b)^x \):
\( 35.35 = 50.5(b)^1 \)
Solve for \( b \): \( b = \frac{35.35}{50.5} = 0.7 \)
Check with the next point \( (2, 24.745) \):
\( h(2) = 50.5(0.7)^2 = 50.5(0.49) = 24.745 \), which matches the point. So the function should be \( h(x) = 50.5(0.7)^x \) (assuming the other options, not fully shown, but based on the given option A: \( h(x)=10.605(50.5)^x \) is exponential growth, which doesn't match. The correct function is exponential decay with \( a = 50.5 \) and \( b = 0.7 \), so option A is incorrect. But since we need to analyze, the process shows the correct model is exponential decay with \( a = 50.5 \) and \( b = 0.7 \).

Answer:

(Assuming the correct option is the one with \( h(x) = 50.5(0.7)^x \), but since only option A is partially shown, if we analyze option A: it has \( a = 10.605 \) and \( b = 50.5 \) (growth), which is wrong. So the correct function (not option A) would be \( h(x) = 50.5(0.7)^x \), but based on the given option A, it's incorrect. However, following the steps, the initial value is 50.5, so the correct function starts with 50.5 as the initial term. )

(Note: Since the full options are not shown, but based on the analysis, the correct function is exponential decay with \( a = 50.5 \) and \( b = 0.7 \), so option A ( \( h(x)=10.605(50.5)^x \) ) is incorrect. If there was an option like \( h(x) = 50.5(0.7)^x \), that would be correct.)