Question
Write the exponential equation shown in the graph (0, 1) and (1, 0.5).
(1 point)
The equation is $$=.
(1 point)
The equation is $$=.
Answers
Answered by
GPT-4o mini
To write the exponential equation based on the given points (0, 1) and (1, 0.5), we can start by using the general form of an exponential function:
\[
y = ab^x
\]
where \(a\) is the y-intercept (value when \(x = 0\)), and \(b\) is the base of the exponential function.
1. **Using the point (0, 1)**:
When \(x = 0\),
\[
y = ab^0 = a \cdot 1 = a
\]
Therefore, from point (0, 1), we have:
\[
a = 1
\]
So our equation simplifies to:
\[
y = b^x
\]
2. **Using the point (1, 0.5)**:
Now, let's use the second point (1, 0.5):
\[
0.5 = b^1
\]
Thus,
\[
b = 0.5
\]
Now we can plug \(a\) and \(b\) back into our equation:
\[
y = 1 \cdot (0.5)^x
\]
or simply:
\[
y = (0.5)^x
\]
Thus, the exponential equation that represents the graph is:
\[
y = (0.5)^x
\]
\[
y = ab^x
\]
where \(a\) is the y-intercept (value when \(x = 0\)), and \(b\) is the base of the exponential function.
1. **Using the point (0, 1)**:
When \(x = 0\),
\[
y = ab^0 = a \cdot 1 = a
\]
Therefore, from point (0, 1), we have:
\[
a = 1
\]
So our equation simplifies to:
\[
y = b^x
\]
2. **Using the point (1, 0.5)**:
Now, let's use the second point (1, 0.5):
\[
0.5 = b^1
\]
Thus,
\[
b = 0.5
\]
Now we can plug \(a\) and \(b\) back into our equation:
\[
y = 1 \cdot (0.5)^x
\]
or simply:
\[
y = (0.5)^x
\]
Thus, the exponential equation that represents the graph is:
\[
y = (0.5)^x
\]
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