Question

a scientist is studying the number of wolves in a forest. the population of wolves in the forest can be modeled by (y = 1240(0.85)^x), where (x) represents the number of years since the study started and (y) represents the number of wolves.
a. determine whether the function represents exponential growth or exponential decay.
b. graph the function.
c. find and interpret the key features of the graph.

Explanation:

Step1: Identify exponential form

The general form of an exponential function is \(y = a(b)^x\), where \(a\) is the initial value and \(b\) is the base. In \(y = 1240(0.85)^x\), \(a = 1240\) and \(b=0.85\). Since \(0\lt b\lt1\), it is exponential decay.

Step2: Create table of values

Choose \(x\) values \(0,1,2,3,4\) and calculate corresponding \(y\) values using \(y = 1240(0.85)^x\).

Step3: Analyze key features

For y - intercept, set \(x = 0\). For asymptote, consider behavior as \(x\to\infty\). Domain is non - negative for years. Range is based on initial value and decay behavior.

Answer:

a. The function represents exponential decay.
b. To graph the function \(y = 1240(0.85)^x\):

• Create a table of values. Choose some values of \(x\) (e.g., \(x = 0,1,2,3,4\)).
• When \(x = 0\), \(y=1240(0.85)^0=1240\).
• When \(x = 1\), \(y = 1240\times0.85 = 1054\).
• When \(x = 2\), \(y=1240\times(0.85)^2=1240\times0.7225 = 895.9\).
• When \(x = 3\), \(y=1240\times(0.85)^3=1240\times0.614125 = 761.515\).
• When \(x = 4\), \(y=1240\times(0.85)^4=1240\times0.52200625 = 647.28775\).
• Plot these points on a coordinate - plane and connect them with a smooth curve. The curve will approach the \(x\) - axis as \(x\) increases.

c. Key features:

• Y - intercept: The y - intercept is at the point \((0,1240)\). This represents the initial population of wolves in the forest when the study started (\(x = 0\) years).
• Asymptote: The \(x\) - axis (\(y = 0\)) is a horizontal asymptote. As \(x\) gets larger and larger (more years pass), the population of wolves gets closer and closer to 0, but never actually reaches 0.
• Domain: The domain is \(x\geq0\) since \(x\) represents the number of years since the study started.
• Range: The range is \(0\lt y\leq1240\) because the population starts at 1240 and decreases towards 0 but never reaches 0.