Answer :
To determine which graph correctly models the given scenario, we need to interpret the equation [tex]\( f(x) = 250(1.10)^{x-4} \)[/tex].
### Explanation
1. Identify the components of the equation:
- [tex]\( f(x) \)[/tex] represents the amount of money in the account.
- [tex]\( 250 \)[/tex] is the initial deposit Akbar made when his son was 4 years old.
- [tex]\( 1.10 \)[/tex] represents the factor by which the initial amount increases each year, due to the 10% annual return.
- [tex]\( (x-4) \)[/tex] represents the number of years since the initial deposit was made; [tex]\( x \)[/tex] is the current age of Akbar's son.
2. Understand the context:
- When [tex]\( x = 4 \)[/tex], the son is 4 years old and the equation should yield the initial deposit, which is [tex]$250. \[ f(4) = 250(1.10)^{4-4} = 250(1.10)^0 = 250 \cdot 1 = 250 \] - As \( x \) increases, that is, as the son gets older, the amount of money will grow due to compound interest. 3. Behavior of the function: - This function represents exponential growth because the base of the exponent (1.10) is greater than 1. - As \( x \) increases, the exponent \( (x - 4) \) also increases, causing the value of \( f(x) \) to grow larger. - For each additional year, the amount of money in the account increases by 10% of the amount from the previous year. 4. Graph characteristics: - At \( x = 4 \), the graph should start at $[/tex]250.
- The graph should exhibit exponential growth, meaning it starts slow but increases more rapidly as [tex]\( x \)[/tex] becomes larger.
- The nature of exponential functions indicates that [tex]\( f(x) \)[/tex] never decreases; it continues to rise as [tex]\( x \)[/tex] increases.
5. Visual representation:
- The correct graph will display an initial value of [tex]$250 at \( x = 4 \). - From there, the graph will rise smoothly and increasingly steeply, demonstrating the compounding interest effect. - The shape will look like a curve that starts lower on the left and rises more sharply on the right. ### Analyzing Options (if any provided): - Incorrect Graphs: - Any graph that does not start at $[/tex]250 when [tex]\( x = 4 \)[/tex] is incorrect.
- Any graph that decreases at any point is incorrect.
- Correct Graph:
- Starts at [tex]$250 when \( x = 4 \). - Shows exponential growth as \( x \) increases beyond 4. Thus, the correct graph is the one that starts at $[/tex]250 at the age of 4 and grows exponentially as the age increases, clearly modeling the scenario as described by the equation [tex]\( f(x) = 250(1.10)^{x-4} \)[/tex].
By matching these criteria to the provided graphs, you can identify which one accurately depicts this financial growth scenario.
### Explanation
1. Identify the components of the equation:
- [tex]\( f(x) \)[/tex] represents the amount of money in the account.
- [tex]\( 250 \)[/tex] is the initial deposit Akbar made when his son was 4 years old.
- [tex]\( 1.10 \)[/tex] represents the factor by which the initial amount increases each year, due to the 10% annual return.
- [tex]\( (x-4) \)[/tex] represents the number of years since the initial deposit was made; [tex]\( x \)[/tex] is the current age of Akbar's son.
2. Understand the context:
- When [tex]\( x = 4 \)[/tex], the son is 4 years old and the equation should yield the initial deposit, which is [tex]$250. \[ f(4) = 250(1.10)^{4-4} = 250(1.10)^0 = 250 \cdot 1 = 250 \] - As \( x \) increases, that is, as the son gets older, the amount of money will grow due to compound interest. 3. Behavior of the function: - This function represents exponential growth because the base of the exponent (1.10) is greater than 1. - As \( x \) increases, the exponent \( (x - 4) \) also increases, causing the value of \( f(x) \) to grow larger. - For each additional year, the amount of money in the account increases by 10% of the amount from the previous year. 4. Graph characteristics: - At \( x = 4 \), the graph should start at $[/tex]250.
- The graph should exhibit exponential growth, meaning it starts slow but increases more rapidly as [tex]\( x \)[/tex] becomes larger.
- The nature of exponential functions indicates that [tex]\( f(x) \)[/tex] never decreases; it continues to rise as [tex]\( x \)[/tex] increases.
5. Visual representation:
- The correct graph will display an initial value of [tex]$250 at \( x = 4 \). - From there, the graph will rise smoothly and increasingly steeply, demonstrating the compounding interest effect. - The shape will look like a curve that starts lower on the left and rises more sharply on the right. ### Analyzing Options (if any provided): - Incorrect Graphs: - Any graph that does not start at $[/tex]250 when [tex]\( x = 4 \)[/tex] is incorrect.
- Any graph that decreases at any point is incorrect.
- Correct Graph:
- Starts at [tex]$250 when \( x = 4 \). - Shows exponential growth as \( x \) increases beyond 4. Thus, the correct graph is the one that starts at $[/tex]250 at the age of 4 and grows exponentially as the age increases, clearly modeling the scenario as described by the equation [tex]\( f(x) = 250(1.10)^{x-4} \)[/tex].
By matching these criteria to the provided graphs, you can identify which one accurately depicts this financial growth scenario.
