Question
On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function goes through points (negative 2, two-ninths), (negative 1, two-thirds), (0, 2), (1, 6), (2, 18), and (3, 54).What is the multiplicative rate of change for the exponential function graphed to the left?
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Answer
Which graph represents the function f(x) =three-halves(2)x? 1. On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function crosses the y-axis at (0, 1) and goes through (1, 3), (2, 9). 2. On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function crosses the y-axis at (0, 2) and goes through (1, 3), (2, 4.5), (3, 7) 3. On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function crosses the y-axis at (0, 1) and goes through (1, 2), (2, 4), (3, 8). 4. On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function crosses the y-axis at (0, 1.5) and goes through (1, 3), (2, 6).
Answered by
GPT-4o mini
To determine the multiplicative rate of change of the exponential function, we can analyze the coordinates provided. The points where the function passes through are:
- \((-2, \frac{2}{9})\)
- \((-1, \frac{2}{3})\)
- \((0, 2)\)
- \((1, 6)\)
- \((2, 18)\)
- \((3, 54)\)
An exponential function can generally be expressed in the form \(y = ab^x\) where \(a\) is the initial value when \(x=0\) and \(b\) is the base that represents the multiplicative rate of change.
From the given points, we can see the function value when \(x = 0\) is \(y = 2\), suggesting that \(a = 2\). Therefore, we can express the function as:
\[
y = 2b^x
\]
Next, we will determine the base \(b\) by looking at the values of \(y\) when \(x\) increases by 1.
Using the given points:
- From \(x = 0\) to \(x = 1\):
\[
\frac{y(1)}{y(0)} = \frac{6}{2} = 3
\]
This means when \(x\) increases by 1, \(y\) is multiplied by 3.
- From \(x = 1\) to \(x = 2\):
\[
\frac{y(2)}{y(1)} = \frac{18}{6} = 3
\]
Again, \(y\) is multiplied by 3.
- From \(x = 2\) to \(x = 3\):
\[
\frac{y(3)}{y(2)} = \frac{54}{18} = 3
\]
This confirms that for each step of \(x\) increasing by 1, the value of \(y\) is consistently multiplied by 3.
Therefore, we conclude that the multiplicative rate of change, denoted as \(b\), for the exponential function is:
\[
\boxed{3}
\]
- \((-2, \frac{2}{9})\)
- \((-1, \frac{2}{3})\)
- \((0, 2)\)
- \((1, 6)\)
- \((2, 18)\)
- \((3, 54)\)
An exponential function can generally be expressed in the form \(y = ab^x\) where \(a\) is the initial value when \(x=0\) and \(b\) is the base that represents the multiplicative rate of change.
From the given points, we can see the function value when \(x = 0\) is \(y = 2\), suggesting that \(a = 2\). Therefore, we can express the function as:
\[
y = 2b^x
\]
Next, we will determine the base \(b\) by looking at the values of \(y\) when \(x\) increases by 1.
Using the given points:
- From \(x = 0\) to \(x = 1\):
\[
\frac{y(1)}{y(0)} = \frac{6}{2} = 3
\]
This means when \(x\) increases by 1, \(y\) is multiplied by 3.
- From \(x = 1\) to \(x = 2\):
\[
\frac{y(2)}{y(1)} = \frac{18}{6} = 3
\]
Again, \(y\) is multiplied by 3.
- From \(x = 2\) to \(x = 3\):
\[
\frac{y(3)}{y(2)} = \frac{54}{18} = 3
\]
This confirms that for each step of \(x\) increasing by 1, the value of \(y\) is consistently multiplied by 3.
Therefore, we conclude that the multiplicative rate of change, denoted as \(b\), for the exponential function is:
\[
\boxed{3}
\]
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