Question

mathematics ii a thayer fall 25 analyzing functional relationships a county commits to decreasing spending for infrastructure in various ways at a rate of 30% per year. at the time of the announcement, the county is spending $12 billion per year. which graph models the amount of infrastructure spending for future years?

Explanation:

Step1: Identify the type of function

The county is decreasing spending at a rate of 30% per year, so this is an exponential decay function. The general form of an exponential decay function is \( y = a(1 - r)^t \), where \( a \) is the initial amount, \( r \) is the rate of decay, and \( t \) is time. Here, \( a = 12 \) billion and \( r = 0.3 \), so the function is \( y = 12(0.7)^t \).

Step2: Analyze the graph features

• Exponential decay graphs are decreasing, concave up, and have a horizontal asymptote at \( y = 0 \) (since the spending can't be negative).
• At \( t = 0 \) (initial time), \( y = 12 \), so the y - intercept should be 12.

Now let's analyze the given graphs:

• The first graph: Check the y - intercept. If we assume the y - axis is for spending (in billions) and x - axis for years. The first graph has a curve that starts at a y - value (let's assume the y - intercept) and decreases. Wait, maybe we can check the point. Let's see, for exponential decay, when \( t = 0 \), \( y = 12 \). Let's check the second graph: Wait, maybe the first graph (top - most) has a y - intercept around 12? Wait, maybe the first graph (the top one) has a curve that starts at (0,12) and decreases, which is consistent with exponential decay. The other graphs: The third graph is increasing (since it goes up as x increases), so it's out. The fourth graph is also increasing. The second graph: Let's check the y - intercept. If the second graph has a y - intercept less than 12, it's out. The first graph (top) has a y - intercept around 12 (since at \( t = 0 \), \( y = 12 \)) and it's a decreasing curve, which matches exponential decay.

Answer:

The top - most graph (the first one among the four) models the amount of infrastructure spending. (Assuming the first graph has a y - intercept of 12 and is a decreasing exponential curve)