Answer :
To determine whether the equation [tex]\(y = 20000(0.95)^x\)[/tex] represents growth or decay, we can analyze the components of the equation. This is an example of an exponential function, which is generally written in the form [tex]\(y = a(b)^x\)[/tex].
1. Identify the initial value and base:
- Here, [tex]\(a = 20000\)[/tex] is the initial value. It indicates the starting amount when [tex]\(x = 0\)[/tex].
- The base [tex]\(b = 0.95\)[/tex] is the crucial part to determine whether the function represents growth or decay.
2. Determine if the base indicates growth or decay:
- If the base [tex]\(b\)[/tex] is greater than 1 ([tex]\(b > 1\)[/tex]), the function represents growth. This means that as [tex]\(x\)[/tex] increases, the value of [tex]\(y\)[/tex] will also increase.
- If the base [tex]\(b\)[/tex] is between 0 and 1 ([tex]\(0 < b < 1\)[/tex]), the function represents decay. This indicates that as [tex]\(x\)[/tex] increases, the value of [tex]\(y\)[/tex] decreases.
3. Apply the criteria to the given base:
- In this equation, [tex]\(b = 0.95\)[/tex] which falls between 0 and 1.
Since the base is 0.95, which is less than 1, the function represents decay. The value of [tex]\(y\)[/tex] will decrease as [tex]\(x\)[/tex] increases.
1. Identify the initial value and base:
- Here, [tex]\(a = 20000\)[/tex] is the initial value. It indicates the starting amount when [tex]\(x = 0\)[/tex].
- The base [tex]\(b = 0.95\)[/tex] is the crucial part to determine whether the function represents growth or decay.
2. Determine if the base indicates growth or decay:
- If the base [tex]\(b\)[/tex] is greater than 1 ([tex]\(b > 1\)[/tex]), the function represents growth. This means that as [tex]\(x\)[/tex] increases, the value of [tex]\(y\)[/tex] will also increase.
- If the base [tex]\(b\)[/tex] is between 0 and 1 ([tex]\(0 < b < 1\)[/tex]), the function represents decay. This indicates that as [tex]\(x\)[/tex] increases, the value of [tex]\(y\)[/tex] decreases.
3. Apply the criteria to the given base:
- In this equation, [tex]\(b = 0.95\)[/tex] which falls between 0 and 1.
Since the base is 0.95, which is less than 1, the function represents decay. The value of [tex]\(y\)[/tex] will decrease as [tex]\(x\)[/tex] increases.
