Question

write a recursive rule for the exponential function. (see example 3)

1. \\( y = 19(4)^t

problem solving

1. a city has a population of 25,000. the population is expected to increase by 5.5% annually for the next decade. (see example 5)

a. write a function that represents the population y after t years.
b. graph the function from part (a). use the graph to estimate the population after 4 years.
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$$\begin{tabular}{|c|} \\hline \\textbf{city population} \\\\ \\begin{tikzpicture}scale=0.6 \\draw-> (0,0) -- (9,0) noderight {$t$}; \\draw-> (0,0) -- (0,45000) nodeabove {population}; \\foreach \\x in {0,1,2,3,4,5,6,7,8} \\draw (\\x,0.1) -- (\\x,-0.1) nodebelow {\\x}; \\foreach \\y in {5000,10000,15000,20000,25000,30000,35000,40000} \\draw (0.1,\\y) -- (-0.1,\\y) nodeleft {\\y}; \\draw (0,25000) -- (8,25000*1.055^8); % sample exponential curve \\filldrawblack (4,25000*1.055^4) circle (2pt); % point at t=4 \\end{tikzpicture} \\\\ \\hline \\end{tabular}$$

Explanation:

Problem 28: Write a recursive rule for the exponential function \( y = 19(4)^t \)

Step 1: Recall the form of exponential functions

An exponential function in the form \( y = a(b)^t \) has a recursive rule. For an exponential function, the recursive rule relates the next term to the previous term. The general form of a recursive rule for an exponential function \( y_t = a(b)^t \) is \( y_0 = a \) (the initial value) and \( y_{t} = b \cdot y_{t - 1} \) for \( t \geq 1 \).

Step 2: Identify \( a \) and \( b \) from the given function

In the function \( y = 19(4)^t \), the initial value \( a \) (when \( t = 0 \)) is \( 19 \) because when \( t = 0 \), \( y = 19(4)^0=19\times1 = 19 \). The base \( b \) of the exponential function is \( 4 \).

Step 3: Write the recursive rule

The initial term (when \( t = 0 \)) is \( y_0=19 \). For each subsequent term (when \( t\geq1 \)), the value of \( y_t \) is \( 4 \) times the previous term \( y_{t - 1} \). So the recursive rule is:

• \( y_0 = 19 \)
• \( y_t=4\cdot y_{t - 1} \) for \( t\geq1 \)

Step 1: Recall the formula for exponential growth

The formula for exponential growth is \( y=a(1 + r)^t \), where \( a \) is the initial amount, \( r \) is the annual growth rate (expressed as a decimal), and \( t \) is the time in years.

Step 2: Identify \( a \) and \( r \)

The initial population \( a = 25000 \). The annual growth rate \( r=5.5\%=0.055 \).

Step 3: Substitute \( a \) and \( r \) into the formula

Substituting \( a = 25000 \) and \( r = 0.055 \) into the exponential growth formula \( y=a(1 + r)^t \), we get:
\( y = 25000(1 + 0.055)^t=25000(1.055)^t \)

Step 1: Analyze the function \( y = 25000(1.055)^t \)

We can find some key points to graph the function:

• When \( t = 0 \), \( y=25000(1.055)^0 = 25000 \)
• When \( t = 1 \), \( y=25000(1.055)^1=25000\times1.055 = 26375 \)
• When \( t = 2 \), \( y=25000(1.055)^2=25000\times(1.055)^2\approx25000\times1.113025 = 27825.625 \)
• When \( t = 3 \), \( y=25000(1.055)^3\approx25000\times1.174241375 = 29356.034375 \)
• When \( t = 4 \), \( y=25000(1.055)^4\approx25000\times1.238824650625 = 30970.616265625 \)

Step 2: Graph the function (brief description)

To graph \( y = 25000(1.055)^t \), we plot the points \((0, 25000)\), \((1, 26375)\), \((2, 27825.63)\), \((3, 29356.03)\), \((4, 30970.62)\) etc. on the given grid (with \( t \) on the x - axis and population on the y - axis).

Step 3: Estimate the population after 4 years from the graph

Looking at the graph, when \( t = 4 \), we can estimate the population. From the calculation above, the exact value is approximately \( 30971 \) (or we can estimate from the graph by finding the point where \( t = 4 \) and reading the y - value. Since the graph has a grid with y - axis marks at 5000, 10000, 15000, 20000, 25000, 30000, 35000, 40000, at \( t = 4 \), the point should be a little above 30000, which matches our calculated value of approximately 31000 (or more precisely around 30970).

Answer:

The recursive rule is \( y_0 = 19 \) and \( y_t = 4y_{t - 1} \) for \( t\geq1 \)

Problem 65a: Write a function that represents the population \( y \) after \( t \) years