Answer :
Certainly! Let's analyze the logistic function given, which is:
[tex]\[y = \frac{89.442}{1 + 4.6342 e^{-0.7813 x}}\][/tex]
The function models the cumulative percent of boys between ages 15 and 20 who have been sexually active at some time, where [tex]\(x\)[/tex] represents the number of years after age 15. Let's go through the steps to solve parts (a) through (d).
### (a) Graph the function for [tex]\(0 \leq x \leq 5\)[/tex]
To properly address part (a), we need to plot the function for [tex]\(x\)[/tex] values in the interval from 0 to 5. Since we can't graph here directly, I'll describe what the graph should look like and provide some key points that you can use to verify the answer:
1. At [tex]\(x = 0\)[/tex]:
[tex]\[ y = \frac{89.442}{1 + 4.6342 e^0} = \frac{89.442}{1 + 4.6342} = \frac{89.442}{5.6342} \approx 15.871 \][/tex]
So, the point [tex]\((0, 15.871)\)[/tex] is on the graph.
2. As [tex]\(x\)[/tex] increases:
- The exponent [tex]\(-0.7813 x\)[/tex] means that the value [tex]\(4.6342 e^{-0.7813 x}\)[/tex] will decrease as [tex]\(x\)[/tex] increases, making the denominator smaller.
- This causes the value of [tex]\(y\)[/tex] to increase and approach the asymptotic value of [tex]\(y = 89.442\)[/tex].
3. Behavior:
- Initially ([tex]\(x = 0\)[/tex]), the function value is around 15.871.
- For intermediate values of [tex]\(x\)[/tex], the function values start to increase rapidly.
- As [tex]\(x\)[/tex] approaches 5, the values get closer to the upper limit, 89.442.
Given this trend, the correct graph will start close to 15.871 at [tex]\(x = 0\)[/tex] and asymptotically approach 89.442 as [tex]\(x\)[/tex] approaches 5.
### Choosing the correct graph:
Without visuals, let’s put it together verbally. The correct graph will:
- Start around 15.871 at [tex]\(x = 0\)[/tex]
- Increase steadily as [tex]\(x\)[/tex] increases but more sharply initially before leveling off
- Approach but not exceed 89.442 as [tex]\(x\)[/tex] reaches 5
Compare this description with the provided options to choose the graph that matches.
### (b), (c), (d) Analysis
As these sub-questions are not detailed in your initial message, assuming they ask for further analysis such as finding specific [tex]\(y\)[/tex] values for certain [tex]\(x\)[/tex] input, or properties about the rate of change within given intervals, a similar approach breakdown and use of function properties and trends would be used.
For detailed sub-parts:
- b & c) calculating derivatives if analyzing the growth rate.
- d) finding specific intercepts by plugging given [tex]\(x\)[/tex] or [tex]\(y\)[/tex] values into the general equation.
Always ensure to return to the core logistic properties indicating initial slow start, rapid middle increase, and a plateau near the carrying capacity (here indicated by the numerator 89.442).
In this breakdown structure, it will facilitate clear logical steps and interpretation suitable for even advanced learners requiring contextually sound mathematical explanation.
[tex]\[y = \frac{89.442}{1 + 4.6342 e^{-0.7813 x}}\][/tex]
The function models the cumulative percent of boys between ages 15 and 20 who have been sexually active at some time, where [tex]\(x\)[/tex] represents the number of years after age 15. Let's go through the steps to solve parts (a) through (d).
### (a) Graph the function for [tex]\(0 \leq x \leq 5\)[/tex]
To properly address part (a), we need to plot the function for [tex]\(x\)[/tex] values in the interval from 0 to 5. Since we can't graph here directly, I'll describe what the graph should look like and provide some key points that you can use to verify the answer:
1. At [tex]\(x = 0\)[/tex]:
[tex]\[ y = \frac{89.442}{1 + 4.6342 e^0} = \frac{89.442}{1 + 4.6342} = \frac{89.442}{5.6342} \approx 15.871 \][/tex]
So, the point [tex]\((0, 15.871)\)[/tex] is on the graph.
2. As [tex]\(x\)[/tex] increases:
- The exponent [tex]\(-0.7813 x\)[/tex] means that the value [tex]\(4.6342 e^{-0.7813 x}\)[/tex] will decrease as [tex]\(x\)[/tex] increases, making the denominator smaller.
- This causes the value of [tex]\(y\)[/tex] to increase and approach the asymptotic value of [tex]\(y = 89.442\)[/tex].
3. Behavior:
- Initially ([tex]\(x = 0\)[/tex]), the function value is around 15.871.
- For intermediate values of [tex]\(x\)[/tex], the function values start to increase rapidly.
- As [tex]\(x\)[/tex] approaches 5, the values get closer to the upper limit, 89.442.
Given this trend, the correct graph will start close to 15.871 at [tex]\(x = 0\)[/tex] and asymptotically approach 89.442 as [tex]\(x\)[/tex] approaches 5.
### Choosing the correct graph:
Without visuals, let’s put it together verbally. The correct graph will:
- Start around 15.871 at [tex]\(x = 0\)[/tex]
- Increase steadily as [tex]\(x\)[/tex] increases but more sharply initially before leveling off
- Approach but not exceed 89.442 as [tex]\(x\)[/tex] reaches 5
Compare this description with the provided options to choose the graph that matches.
### (b), (c), (d) Analysis
As these sub-questions are not detailed in your initial message, assuming they ask for further analysis such as finding specific [tex]\(y\)[/tex] values for certain [tex]\(x\)[/tex] input, or properties about the rate of change within given intervals, a similar approach breakdown and use of function properties and trends would be used.
For detailed sub-parts:
- b & c) calculating derivatives if analyzing the growth rate.
- d) finding specific intercepts by plugging given [tex]\(x\)[/tex] or [tex]\(y\)[/tex] values into the general equation.
Always ensure to return to the core logistic properties indicating initial slow start, rapid middle increase, and a plateau near the carrying capacity (here indicated by the numerator 89.442).
In this breakdown structure, it will facilitate clear logical steps and interpretation suitable for even advanced learners requiring contextually sound mathematical explanation.
