Question
Find the equation of the exponential function represented by the table below:
xx yy
00 0, point, 20.2
11 0, point, 10.1
22 0, point, 0, 50.05
33 0, point, 0, 2, 50.025
xx yy
00 0, point, 20.2
11 0, point, 10.1
22 0, point, 0, 50.05
33 0, point, 0, 2, 50.025
Answers
GPT-4o mini
To find the equation of the exponential function represented by the table, let's interpret the data correctly. Given the values:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 0.2 \\
1 & 0.1 \\
2 & 0.05 \\
3 & 0.025 \\
\hline
\end{array}
\]
We can see that as \( x \) increases, \( y \) decreases, suggesting an exponential decay function. The general form of an exponential function is:
\[
y = ab^x
\]
where \( a \) is the initial value (the value of \( y \) when \( x = 0 \)), and \( b \) is the base of the exponential function.
From the table, we see that when \( x = 0 \), \( y = 0.2 \). Therefore, \( a = 0.2 \).
Next, we need to determine the base \( b \). We can use one of the other points to set up an equation. We can use the point \( (1, 0.1) \):
\[
0.1 = 0.2b^1
\]
Simplifying:
\[
b = \frac{0.1}{0.2} = 0.5
\]
Now let's verify that this \( b \) value holds for the other points in the table.
For \( x = 2 \):
\[
y = 0.2(0.5)^2 = 0.2 \times 0.25 = 0.05
\]
For \( x = 3 \):
\[
y = 0.2(0.5)^3 = 0.2 \times 0.125 = 0.025
\]
Since the values match the table, we have verified our function.
Therefore, the equation of the exponential function represented by the table is:
\[
y = 0.2(0.5)^x
\]
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 0.2 \\
1 & 0.1 \\
2 & 0.05 \\
3 & 0.025 \\
\hline
\end{array}
\]
We can see that as \( x \) increases, \( y \) decreases, suggesting an exponential decay function. The general form of an exponential function is:
\[
y = ab^x
\]
where \( a \) is the initial value (the value of \( y \) when \( x = 0 \)), and \( b \) is the base of the exponential function.
From the table, we see that when \( x = 0 \), \( y = 0.2 \). Therefore, \( a = 0.2 \).
Next, we need to determine the base \( b \). We can use one of the other points to set up an equation. We can use the point \( (1, 0.1) \):
\[
0.1 = 0.2b^1
\]
Simplifying:
\[
b = \frac{0.1}{0.2} = 0.5
\]
Now let's verify that this \( b \) value holds for the other points in the table.
For \( x = 2 \):
\[
y = 0.2(0.5)^2 = 0.2 \times 0.25 = 0.05
\]
For \( x = 3 \):
\[
y = 0.2(0.5)^3 = 0.2 \times 0.125 = 0.025
\]
Since the values match the table, we have verified our function.
Therefore, the equation of the exponential function represented by the table is:
\[
y = 0.2(0.5)^x
\]