Given the graph of the exponential equation, which option is the correct exponential equation for the graph?
the points are the graph are ( 1,14) and (2,4)
Option #1: y=7(12)x
Option #2: y=49(27)x
Option #3: y=7(27)x
Option #4: y=49(72)x
(1 point)
Option #
is the correct equa
the points are the graph are ( 1,14) and (2,4)
Option #1: y=7(12)x
Option #2: y=49(27)x
Option #3: y=7(27)x
Option #4: y=49(72)x
(1 point)
Option #
is the correct equa
Answers
Answered by
GPT-4o mini
To determine which option represents the correct exponential equation given the points (1, 14) and (2, 4), we need to derive the exponential equation of the form \( y = ab^x \).
1. We'll start with the general form:
\[ y = ab^x \]
2. Using the point (1, 14):
\[ 14 = ab^1 \]
This simplifies to:
\[ ab = 14 \quad \text{(1)} \]
3. Using the point (2, 4):
\[ 4 = ab^2 \]
This simplifies to:
\[ ab^2 = 4 \quad \text{(2)} \]
4. From equation (1), we can express \( a \) in terms of \( b \):
\[ a = \frac{14}{b} \]
5. Substitute \( a \) into equation (2):
\[ \frac{14}{b}b^2 = 4 \]
Simplifying this gives:
\[ 14b = 4 \]
\[ b = \frac{4}{14} = \frac{2}{7} \]
6. Now, substitute \( b = \frac{2}{7} \) back into equation (1) to find \( a \):
\[ a \left( \frac{2}{7} \right) = 14 \]
\[ a = 14 \cdot \frac{7}{2} = 49 \]
So the equation we've derived is:
\[ y = 49 \left( \frac{2}{7} \right)^x \]
Now, let's analyze the options provided:
1. Option #1: \( y = 7(12)^x \) - Doesn't match.
2. Option #2: \( y = 49(27)^x \) - Doesn't match.
3. Option #3: \( y = 7(27)^x \) - Doesn't match.
4. Option #4: \( y = 49(72)^x \) - Doesn't match.
None of the options provided match the derived equation \( y = 49 \left( \frac{2}{7} \right)^x \). It appears there may be a misunderstanding in the problem setup based on the provided options.
To check if the equations were miscommunicated or values errored, ensure the relationships hold for the points given.
If you wanted the closest form with the correct base or if you've got different points or conclusions in consideration based on the exponential growth parameters, please clarify accordingly.
1. We'll start with the general form:
\[ y = ab^x \]
2. Using the point (1, 14):
\[ 14 = ab^1 \]
This simplifies to:
\[ ab = 14 \quad \text{(1)} \]
3. Using the point (2, 4):
\[ 4 = ab^2 \]
This simplifies to:
\[ ab^2 = 4 \quad \text{(2)} \]
4. From equation (1), we can express \( a \) in terms of \( b \):
\[ a = \frac{14}{b} \]
5. Substitute \( a \) into equation (2):
\[ \frac{14}{b}b^2 = 4 \]
Simplifying this gives:
\[ 14b = 4 \]
\[ b = \frac{4}{14} = \frac{2}{7} \]
6. Now, substitute \( b = \frac{2}{7} \) back into equation (1) to find \( a \):
\[ a \left( \frac{2}{7} \right) = 14 \]
\[ a = 14 \cdot \frac{7}{2} = 49 \]
So the equation we've derived is:
\[ y = 49 \left( \frac{2}{7} \right)^x \]
Now, let's analyze the options provided:
1. Option #1: \( y = 7(12)^x \) - Doesn't match.
2. Option #2: \( y = 49(27)^x \) - Doesn't match.
3. Option #3: \( y = 7(27)^x \) - Doesn't match.
4. Option #4: \( y = 49(72)^x \) - Doesn't match.
None of the options provided match the derived equation \( y = 49 \left( \frac{2}{7} \right)^x \). It appears there may be a misunderstanding in the problem setup based on the provided options.
To check if the equations were miscommunicated or values errored, ensure the relationships hold for the points given.
If you wanted the closest form with the correct base or if you've got different points or conclusions in consideration based on the exponential growth parameters, please clarify accordingly.