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To determine the best description of the graph of the function [tex]\( f(x) = 60 \left( \frac{1}{3} \right)^x \)[/tex], let's break down the components of this exponential function.
### Step-by-Step Analysis:
1. Initial Value of the Function:
- The general form of an exponential function is [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is the initial value when [tex]\( x = 0 \)[/tex].
- Here, [tex]\( a = 60 \)[/tex]. Therefore, the initial value of the function when [tex]\( x = 0 \)[/tex] is [tex]\( f(0) = 60 \cdot \left( \frac{1}{3} \right)^0 = 60 \)[/tex].
2. Successive Term Determination:
- The base [tex]\( b \)[/tex] in the function [tex]\( f(x) \)[/tex] determines how each successive term is derived from the previous term.
- Here, [tex]\( b = \frac{1}{3} \)[/tex]. Each successive term is obtained by multiplying the previous term by [tex]\( \frac{1}{3} \)[/tex].
### Conclusion:
With the initial value and successive term determined, we compare these findings against the descriptions given in the options:
1. Option 1: "The graph has an initial value of 20, and each successive term is determined by subtracting [tex]\( \frac{1}{3} \)[/tex]."
- This is incorrect because the initial value is not 20, and the terms are not determined by subtraction.
2. Option 2: "The graph has an initial value of 20, and each successive term is determined by multiplying by [tex]\( \frac{1}{3} \)[/tex]."
- This is incorrect because the initial value is not 20.
3. Option 3: "The graph has an initial value of 60, and each successive term is determined by subtracting [tex]\( \frac{1}{3} \)[/tex]."
- This is incorrect because the terms are not determined by subtraction.
4. Option 4: "The graph has an initial value of 60, and each successive term is determined by multiplying by [tex]\( \frac{1}{3} \)[/tex]."
- This is correct because the initial value is 60, and each successive term is determined by multiplying the previous term by [tex]\( \frac{1}{3} \)[/tex].
### Final Answer:
The best description of the graph of the function [tex]\( f(x) = 60 \left( \frac{1}{3} \right)^x \)[/tex] is:
"The graph has an initial value of 60, and each successive term is determined by multiplying by [tex]\( \frac{1}{3} \)[/tex]."
### Step-by-Step Analysis:
1. Initial Value of the Function:
- The general form of an exponential function is [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is the initial value when [tex]\( x = 0 \)[/tex].
- Here, [tex]\( a = 60 \)[/tex]. Therefore, the initial value of the function when [tex]\( x = 0 \)[/tex] is [tex]\( f(0) = 60 \cdot \left( \frac{1}{3} \right)^0 = 60 \)[/tex].
2. Successive Term Determination:
- The base [tex]\( b \)[/tex] in the function [tex]\( f(x) \)[/tex] determines how each successive term is derived from the previous term.
- Here, [tex]\( b = \frac{1}{3} \)[/tex]. Each successive term is obtained by multiplying the previous term by [tex]\( \frac{1}{3} \)[/tex].
### Conclusion:
With the initial value and successive term determined, we compare these findings against the descriptions given in the options:
1. Option 1: "The graph has an initial value of 20, and each successive term is determined by subtracting [tex]\( \frac{1}{3} \)[/tex]."
- This is incorrect because the initial value is not 20, and the terms are not determined by subtraction.
2. Option 2: "The graph has an initial value of 20, and each successive term is determined by multiplying by [tex]\( \frac{1}{3} \)[/tex]."
- This is incorrect because the initial value is not 20.
3. Option 3: "The graph has an initial value of 60, and each successive term is determined by subtracting [tex]\( \frac{1}{3} \)[/tex]."
- This is incorrect because the terms are not determined by subtraction.
4. Option 4: "The graph has an initial value of 60, and each successive term is determined by multiplying by [tex]\( \frac{1}{3} \)[/tex]."
- This is correct because the initial value is 60, and each successive term is determined by multiplying the previous term by [tex]\( \frac{1}{3} \)[/tex].
### Final Answer:
The best description of the graph of the function [tex]\( f(x) = 60 \left( \frac{1}{3} \right)^x \)[/tex] is:
"The graph has an initial value of 60, and each successive term is determined by multiplying by [tex]\( \frac{1}{3} \)[/tex]."
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