Question
Given the graph of the exponential function, which option is the correct exponential equation for the graph? (The first point is at (3, 9), the second point is at (4, 6)
y = 3(1/2)^x
y = 3(2/4)^x
y = 30(2/3)^x
y= 30(3/2)^x
y = 3(1/2)^x
y = 3(2/4)^x
y = 30(2/3)^x
y= 30(3/2)^x
Answers
Answered by
GPT-4o mini
To identify the correct exponential equation from the options provided based on the points (3, 9) and (4, 6), let's analyze the information systematically.
Given two points on the graph:
1. \( (3, 9) \)
2. \( (4, 6) \)
This indicates that the function should satisfy the following conditions:
- For \( x = 3 \), \( y = 9 \)
- For \( x = 4 \), \( y = 6 \)
Let's test each of the equations one by one:
1. **y = 3(1/2)^x**
For \( x = 3 \):
\[
y = 3(1/2)^3 = 3 \cdot \frac{1}{8} = \frac{3}{8} \quad \text{(not 9)}
\]
For \( x = 4 \):
\[
y = 3(1/2)^4 = 3 \cdot \frac{1}{16} = \frac{3}{16} \quad \text{(not 6)}
\]
2. **y = 3(2/4)^x**
This simplifies to:
\[
y = 3(1/2)^x
\]
This is the same as the first option, which we've already determined does not work.
3. **y = 30(2/3)^x**
For \( x = 3 \):
\[
y = 30(2/3)^3 = 30 \cdot \frac{8}{27} = \frac{240}{27} \approx 8.89 \quad \text{(not 9)}
\]
For \( x = 4 \):
\[
y = 30(2/3)^4 = 30 \cdot \frac{16}{81} = \frac{480}{81} \approx 5.93 \quad \text{(not 6)}
\]
4. **y = 30(3/2)^x**
For \( x = 3 \):
\[
y = 30(3/2)^3 = 30 \cdot \frac{27}{8} = \frac{810}{8} = 101.25 \quad \text{(not 9)}
\]
For \( x = 4 \):
\[
y = 30(3/2)^4 = 30 \cdot \frac{81}{16} = \frac{2430}{16} \approx 151.875 \quad \text{(not 6)}
\]
Since none of the equations perfectly fit the values given, let's take an alternative approach using the slope.
Looking at the points, we see \( y \) is decreasing from \( 9 \) to \( 6 \) when \( x \) increases from \( 3 \) to \( 4 \). To double-check, we'll find the decay factor from the first point to the second point:
- From \( 9 \) at \( x = 3 \) to \( 6 \) at \( x = 4 \):
\[
\text{Decay Factor} = \frac{6}{9} = \frac{2}{3}
\]
This implies the function decreases by a factor of \( \frac{2}{3} \) as \( x \) increases by \( 1 \).
Thus, if the general form \( y = A \left( \frac{2}{3} \right)^{x - B} \), with an initial value of **9** when \( x = 3** and \( Af(3) = 9 \):
- Solving for A using \( B = 3 \) gives a scaled equation \( y = 30(2/3)^{x-3} \).
None perfectly fit as stated, so possibly the graph has misrepresented or misrepresented notes in describing decay.
In perspective:
30(2/3)^x is closest fitting with fluctuations possibly translating values.
The equations need reevaluation with additional points on correct scales for precise chart plotting thus retaining line lines of \( 30(2/3)^x \).
Among the given options, we deduce **y = 30(2/3)^x** embodies complex with visible converging slope checked seems fitting maintaining internal factors seen validating plausible fit model precise scaling.
Given two points on the graph:
1. \( (3, 9) \)
2. \( (4, 6) \)
This indicates that the function should satisfy the following conditions:
- For \( x = 3 \), \( y = 9 \)
- For \( x = 4 \), \( y = 6 \)
Let's test each of the equations one by one:
1. **y = 3(1/2)^x**
For \( x = 3 \):
\[
y = 3(1/2)^3 = 3 \cdot \frac{1}{8} = \frac{3}{8} \quad \text{(not 9)}
\]
For \( x = 4 \):
\[
y = 3(1/2)^4 = 3 \cdot \frac{1}{16} = \frac{3}{16} \quad \text{(not 6)}
\]
2. **y = 3(2/4)^x**
This simplifies to:
\[
y = 3(1/2)^x
\]
This is the same as the first option, which we've already determined does not work.
3. **y = 30(2/3)^x**
For \( x = 3 \):
\[
y = 30(2/3)^3 = 30 \cdot \frac{8}{27} = \frac{240}{27} \approx 8.89 \quad \text{(not 9)}
\]
For \( x = 4 \):
\[
y = 30(2/3)^4 = 30 \cdot \frac{16}{81} = \frac{480}{81} \approx 5.93 \quad \text{(not 6)}
\]
4. **y = 30(3/2)^x**
For \( x = 3 \):
\[
y = 30(3/2)^3 = 30 \cdot \frac{27}{8} = \frac{810}{8} = 101.25 \quad \text{(not 9)}
\]
For \( x = 4 \):
\[
y = 30(3/2)^4 = 30 \cdot \frac{81}{16} = \frac{2430}{16} \approx 151.875 \quad \text{(not 6)}
\]
Since none of the equations perfectly fit the values given, let's take an alternative approach using the slope.
Looking at the points, we see \( y \) is decreasing from \( 9 \) to \( 6 \) when \( x \) increases from \( 3 \) to \( 4 \). To double-check, we'll find the decay factor from the first point to the second point:
- From \( 9 \) at \( x = 3 \) to \( 6 \) at \( x = 4 \):
\[
\text{Decay Factor} = \frac{6}{9} = \frac{2}{3}
\]
This implies the function decreases by a factor of \( \frac{2}{3} \) as \( x \) increases by \( 1 \).
Thus, if the general form \( y = A \left( \frac{2}{3} \right)^{x - B} \), with an initial value of **9** when \( x = 3** and \( Af(3) = 9 \):
- Solving for A using \( B = 3 \) gives a scaled equation \( y = 30(2/3)^{x-3} \).
None perfectly fit as stated, so possibly the graph has misrepresented or misrepresented notes in describing decay.
In perspective:
30(2/3)^x is closest fitting with fluctuations possibly translating values.
The equations need reevaluation with additional points on correct scales for precise chart plotting thus retaining line lines of \( 30(2/3)^x \).
Among the given options, we deduce **y = 30(2/3)^x** embodies complex with visible converging slope checked seems fitting maintaining internal factors seen validating plausible fit model precise scaling.