Question
the table represents some points on the graph of an exponential function.
x: 0,1,2,3
f(x): 80,60,45,33.75
f(x): ______
x: 0,1,2,3
f(x): 80,60,45,33.75
f(x): ______
Answers
GPT-4o mini
To find the exponential function that fits the points you've provided, we can start by selecting the general form of an exponential function:
\[ f(x) = a \cdot b^x \]
where \( a \) is the initial value (the value of \( f(0) \)) and \( b \) is the base of the exponential function.
From your data:
- When \( x = 0 \), \( f(0) = 80 \), so \( a = 80 \).
- Using this value, we can find \( b \) using the other points.
Let's calculate \( b \) using the points provided:
1. For \( x = 1 \):
\[ 60 = 80 \cdot b^1 \]
\[ b = \frac{60}{80} = 0.75 \]
Now check if this base \( b \) holds for the remaining points.
2. For \( x = 2 \):
\[ f(2) = 80 \cdot (0.75)^2 = 80 \cdot 0.5625 = 45 \] (This matches!)
3. For \( x = 3 \):
\[ f(3) = 80 \cdot (0.75)^3 = 80 \cdot 0.421875 = 33.75 \] (This also matches!)
Since all points fit well with the computed values, the exponential function that describes the graph of the points is:
\[ f(x) = 80 \cdot (0.75)^x \]
\[ f(x) = a \cdot b^x \]
where \( a \) is the initial value (the value of \( f(0) \)) and \( b \) is the base of the exponential function.
From your data:
- When \( x = 0 \), \( f(0) = 80 \), so \( a = 80 \).
- Using this value, we can find \( b \) using the other points.
Let's calculate \( b \) using the points provided:
1. For \( x = 1 \):
\[ 60 = 80 \cdot b^1 \]
\[ b = \frac{60}{80} = 0.75 \]
Now check if this base \( b \) holds for the remaining points.
2. For \( x = 2 \):
\[ f(2) = 80 \cdot (0.75)^2 = 80 \cdot 0.5625 = 45 \] (This matches!)
3. For \( x = 3 \):
\[ f(3) = 80 \cdot (0.75)^3 = 80 \cdot 0.421875 = 33.75 \] (This also matches!)
Since all points fit well with the computed values, the exponential function that describes the graph of the points is:
\[ f(x) = 80 \cdot (0.75)^x \]