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Sagot :
We are given a table of values for a continuous exponential function [tex]\( f(x) \)[/tex]. Let's start by analyzing the information provided and then use it to graph the function and identify its properties, including the [tex]\( y \)[/tex]-intercept.
### Step-by-Step Solution:
1. Identify the Nature of the Function:
- Since this is an exponential function, [tex]\( f(x) \)[/tex] can be expressed in the general form:
[tex]\[ f(x) = a \cdot r^{x-1} \][/tex]
where [tex]\( a \)[/tex] is a constant and [tex]\( r \)[/tex] is the base of the exponential function.
2. Calculate the Ratio [tex]\( r \)[/tex]:
- To find [tex]\( r \)[/tex], we look at the given consecutive values of [tex]\( f(x) \)[/tex]. For [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
[tex]\[ r = \frac{f(3)}{f(2)} = \frac{3}{9} = \frac{1}{3} \][/tex]
- Check if [tex]\( r = \frac{1}{3} \)[/tex] works for the other values:
[tex]\[ f(4) = f(3) \cdot r = 3 \cdot \frac{1}{3} = 1 \][/tex]
[tex]\[ f(5) = f(4) \cdot r = 1 \cdot \frac{1}{3} = \frac{1}{3} \][/tex]
- Since the ratio is consistent for all given points, we confirm that [tex]\( r = \frac{1}{3} \)[/tex].
3. Determine the Initial Value [tex]\( a \)[/tex]:
- Given [tex]\( f(2) = 9 \)[/tex], we need to express this in terms of [tex]\( a \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ f(2) = a \cdot r^{2-1} = a \cdot r = a \cdot \frac{1}{3} \][/tex]
[tex]\[ 9 = a \cdot \frac{1}{3} \][/tex]
[tex]\[ a = 9 \cdot 3 = 27 \][/tex]
4. Form the Equation of the Function:
- Now that we have [tex]\( a = 27 \)[/tex] and [tex]\( r = \frac{1}{3} \)[/tex], the function [tex]\( f(x) \)[/tex] can be written as:
[tex]\[ f(x) = 27 \cdot \left( \frac{1}{3} \right)^{x-1} \][/tex]
5. Identify the [tex]\( y \)[/tex]-Intercept:
- The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 27 \cdot \left( \frac{1}{3} \right)^{1-1} = 27 \cdot 1 = 27 \][/tex]
6. Graph the Function:
- To graph [tex]\( f(x) \)[/tex], we plot the points from the table and use the function's equation to draw a smooth curve through these points.
Here are the points from the table:
[tex]\[ (2, 9), (3, 3), (4, 1), (5, \frac{1}{3}) \][/tex]
- Additional points might include:
[tex]\[ (1, 27), (0, 81) \text{ (using } f(x) = 27 \cdot \left( \frac{1}{3} \right)^{x-1}) \][/tex]
### Conclusion:
- The [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) \)[/tex] is [tex]\( 27 \)[/tex].
- The graph of [tex]\( f(x) \)[/tex] is an exponential decay curve passing through the points we identified above. Make sure to draw the curve smoothly as per the exponential nature of the function.
### Step-by-Step Solution:
1. Identify the Nature of the Function:
- Since this is an exponential function, [tex]\( f(x) \)[/tex] can be expressed in the general form:
[tex]\[ f(x) = a \cdot r^{x-1} \][/tex]
where [tex]\( a \)[/tex] is a constant and [tex]\( r \)[/tex] is the base of the exponential function.
2. Calculate the Ratio [tex]\( r \)[/tex]:
- To find [tex]\( r \)[/tex], we look at the given consecutive values of [tex]\( f(x) \)[/tex]. For [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
[tex]\[ r = \frac{f(3)}{f(2)} = \frac{3}{9} = \frac{1}{3} \][/tex]
- Check if [tex]\( r = \frac{1}{3} \)[/tex] works for the other values:
[tex]\[ f(4) = f(3) \cdot r = 3 \cdot \frac{1}{3} = 1 \][/tex]
[tex]\[ f(5) = f(4) \cdot r = 1 \cdot \frac{1}{3} = \frac{1}{3} \][/tex]
- Since the ratio is consistent for all given points, we confirm that [tex]\( r = \frac{1}{3} \)[/tex].
3. Determine the Initial Value [tex]\( a \)[/tex]:
- Given [tex]\( f(2) = 9 \)[/tex], we need to express this in terms of [tex]\( a \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ f(2) = a \cdot r^{2-1} = a \cdot r = a \cdot \frac{1}{3} \][/tex]
[tex]\[ 9 = a \cdot \frac{1}{3} \][/tex]
[tex]\[ a = 9 \cdot 3 = 27 \][/tex]
4. Form the Equation of the Function:
- Now that we have [tex]\( a = 27 \)[/tex] and [tex]\( r = \frac{1}{3} \)[/tex], the function [tex]\( f(x) \)[/tex] can be written as:
[tex]\[ f(x) = 27 \cdot \left( \frac{1}{3} \right)^{x-1} \][/tex]
5. Identify the [tex]\( y \)[/tex]-Intercept:
- The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 27 \cdot \left( \frac{1}{3} \right)^{1-1} = 27 \cdot 1 = 27 \][/tex]
6. Graph the Function:
- To graph [tex]\( f(x) \)[/tex], we plot the points from the table and use the function's equation to draw a smooth curve through these points.
Here are the points from the table:
[tex]\[ (2, 9), (3, 3), (4, 1), (5, \frac{1}{3}) \][/tex]
- Additional points might include:
[tex]\[ (1, 27), (0, 81) \text{ (using } f(x) = 27 \cdot \left( \frac{1}{3} \right)^{x-1}) \][/tex]
### Conclusion:
- The [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) \)[/tex] is [tex]\( 27 \)[/tex].
- The graph of [tex]\( f(x) \)[/tex] is an exponential decay curve passing through the points we identified above. Make sure to draw the curve smoothly as per the exponential nature of the function.
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