Question
Find the equation of the exponential function represented by the table below:
xx yy
00 11
11 0, point, 50.5
22 0, point, 2, 50.25
33 0, point, 1, 2, 50.125
xx yy
00 11
11 0, point, 50.5
22 0, point, 2, 50.25
33 0, point, 1, 2, 50.125
Answers
GPT-4o mini
To find the equation of the exponential function that fits the given data points, we first need to organize the information. The points provided can be rewritten in a clearer format:
- \( (0, 1) \)
- \( (1, 0.5) \)
- \( (2, 0.25) \)
- \( (3, 0.125) \)
Assuming an exponential function of the form:
\[
y = ab^x
\]
where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base (the growth/decay factor), we can use the points to find \( a \) and \( b \).
1. **Using the Point (0, 1)**:
When \( x = 0 \):
\[
y = ab^0 = a
\]
This gives \( a = 1 \).
2. **Substituting \( a \) in the general form**:
The function simplifies to:
\[
y = 1 \cdot b^x = b^x
\]
3. **Using the next point (1, 0.5)**:
From the point \( (1, 0.5) \):
\[
0.5 = b^1
\]
Thus, \( b = 0.5 \).
4. **Verifying with other points**:
Now we have \( b \) and can use it to verify with the other points:
- For \( (2, 0.25) \):
\[
y = b^2 = (0.5)^2 = 0.25
\]
- For \( (3, 0.125) \):
\[
y = b^3 = (0.5)^3 = 0.125
\]
Both calculations confirm that our values hold true.
Thus, the equation of the exponential function represented by the table is:
\[
y = (0.5)^x
\]
This is the required exponential function that fits the given data points.
- \( (0, 1) \)
- \( (1, 0.5) \)
- \( (2, 0.25) \)
- \( (3, 0.125) \)
Assuming an exponential function of the form:
\[
y = ab^x
\]
where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base (the growth/decay factor), we can use the points to find \( a \) and \( b \).
1. **Using the Point (0, 1)**:
When \( x = 0 \):
\[
y = ab^0 = a
\]
This gives \( a = 1 \).
2. **Substituting \( a \) in the general form**:
The function simplifies to:
\[
y = 1 \cdot b^x = b^x
\]
3. **Using the next point (1, 0.5)**:
From the point \( (1, 0.5) \):
\[
0.5 = b^1
\]
Thus, \( b = 0.5 \).
4. **Verifying with other points**:
Now we have \( b \) and can use it to verify with the other points:
- For \( (2, 0.25) \):
\[
y = b^2 = (0.5)^2 = 0.25
\]
- For \( (3, 0.125) \):
\[
y = b^3 = (0.5)^3 = 0.125
\]
Both calculations confirm that our values hold true.
Thus, the equation of the exponential function represented by the table is:
\[
y = (0.5)^x
\]
This is the required exponential function that fits the given data points.