Answer :
To find the equation that is equivalent to [tex]\(y = 25000(1.16)^x\)[/tex], we need to understand how negative exponents work and analyze the given options.
1. Understand Negative Exponents:
A negative exponent, like [tex]\((b)^{-x}\)[/tex], is equivalent to [tex]\(\left(\frac{1}{b}\right)^{x}\)[/tex]. This means that a negative exponent turns the base into its reciprocal and changes the sign of the exponent.
2. Analyze each equation:
- Equation 1: [tex]\(y = 25000(1.16)^x\)[/tex]
This is the original equation and is already in the form we are analyzing, so we don't need to change it.
- Equation 2: [tex]\(y = 250000\left(\frac{1}{1.16}\right)^x\)[/tex]
In this case, we have a different scaling factor and a different base for the exponent. According to the properties of exponents, [tex]\(\left(\frac{1}{1.16}\right)^x\)[/tex] suggests an exponential decay scenario, which would normally map to something like [tex]\((1.16)^{-x}\)[/tex]. However, this setup doesn't exactly match [tex]\(y = 25000(1.16)^x\)[/tex] because of both the change in factor and the reciprocal, unless it specifically considers a negative growth model. Here, we should focus on typical equivalences in exponential identities rather than transformations like decay vs. growth.
- Equation 3: [tex]\(y = 25000\left(\frac{1}{16}\right)^x\)[/tex]
This base, [tex]\(\left(\frac{1}{16}\right)^x\)[/tex], is not related to [tex]\(1.16^x\)[/tex]. The base is entirely different and it doesn't represent a simple transformation of [tex]\(1.16^x\)[/tex].
3. Conclusion:
Among the given equations, neither equation directly represents the same growth function as the original without altering function behavior unlike what might be inferred about equivalency regarding scaling and transformation dynamics (inverse representation).
Based on equivalences, the most likely transformation that doesn't change due to a known translated effect seen only in decay scenarios equates to the typical equivalent scenario when transformed purposely.
Therefore, [tex]\(y = 25000\left(\frac{1}{1.16}\right)^{-x}\)[/tex], or equivalently transformed directly as needed without additional changes, makes the difference. This means Equation 2 is the closest in suggestions of traditional interpretations when final choice considerations are checked.
1. Understand Negative Exponents:
A negative exponent, like [tex]\((b)^{-x}\)[/tex], is equivalent to [tex]\(\left(\frac{1}{b}\right)^{x}\)[/tex]. This means that a negative exponent turns the base into its reciprocal and changes the sign of the exponent.
2. Analyze each equation:
- Equation 1: [tex]\(y = 25000(1.16)^x\)[/tex]
This is the original equation and is already in the form we are analyzing, so we don't need to change it.
- Equation 2: [tex]\(y = 250000\left(\frac{1}{1.16}\right)^x\)[/tex]
In this case, we have a different scaling factor and a different base for the exponent. According to the properties of exponents, [tex]\(\left(\frac{1}{1.16}\right)^x\)[/tex] suggests an exponential decay scenario, which would normally map to something like [tex]\((1.16)^{-x}\)[/tex]. However, this setup doesn't exactly match [tex]\(y = 25000(1.16)^x\)[/tex] because of both the change in factor and the reciprocal, unless it specifically considers a negative growth model. Here, we should focus on typical equivalences in exponential identities rather than transformations like decay vs. growth.
- Equation 3: [tex]\(y = 25000\left(\frac{1}{16}\right)^x\)[/tex]
This base, [tex]\(\left(\frac{1}{16}\right)^x\)[/tex], is not related to [tex]\(1.16^x\)[/tex]. The base is entirely different and it doesn't represent a simple transformation of [tex]\(1.16^x\)[/tex].
3. Conclusion:
Among the given equations, neither equation directly represents the same growth function as the original without altering function behavior unlike what might be inferred about equivalency regarding scaling and transformation dynamics (inverse representation).
Based on equivalences, the most likely transformation that doesn't change due to a known translated effect seen only in decay scenarios equates to the typical equivalent scenario when transformed purposely.
Therefore, [tex]\(y = 25000\left(\frac{1}{1.16}\right)^{-x}\)[/tex], or equivalently transformed directly as needed without additional changes, makes the difference. This means Equation 2 is the closest in suggestions of traditional interpretations when final choice considerations are checked.
