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Given the graph of the exponential equation representing the value of a car since purchase, which option is the correct exponen...Question
Given the graph of the exponential equation representing the value of a car since purchase, which option is the correct exponential equation for the graph if the graph goes through the points (0, 3000) and (1, 22500)
a. y=22,500(3/4)^x
b. y=0.75(30,000)^x
c. y=30,000(3/4)^x
d. y=30,000(4/3)^x
a. y=22,500(3/4)^x
b. y=0.75(30,000)^x
c. y=30,000(3/4)^x
d. y=30,000(4/3)^x
Answers
GPT-4o mini
To determine the correct exponential equation from the given options, we will use the points (0, 3000) and (1, 22500) to derive the equation.
The general form of an exponential equation can be represented as:
\[ y = a \cdot b^x \]
Where:
- \( a \) is the initial value (the value at \( x = 0 \)).
- \( b \) is the growth (or decay) factor.
First, we identify the initial value from the point (0, 3000):
- At \( x = 0 \): \( y = 3000 \)
This means \( a = 3000 \).
Next, we can use the point (1, 22500) to find the value of \( b \):
Using the point (1, 22500):
\[ 22500 = 3000 \cdot b^1 \]
To solve for \( b \), we divide both sides by 3000:
\[ b = \frac{22500}{3000} = 7.5 \]
So, we can rewrite the equation as:
\[ y = 3000 \cdot 7.5^x \]
Now we need to check each of the given options and see if any match this relationship, particularly at \( x = 1 \):
1. **Option a: \( y=22,500(3/4)^x \)**
- At \( x = 1 \): \( y = 22,500(3/4) = 22,500 \cdot 0.75 = 16,875 \) (not 22500)
2. **Option b: \( y=0.75(30,000)^x \)**
- At \( x = 1 \): \( y = 0.75 \cdot 30,000 = 22,500 \) (this matches! But the equation at \( x = 0 \) does not match)
3. **Option c: \( y=30,000(3/4)^x \)**
- At \( x = 0 \): \( y = 30,000(1) = 30,000 \) (not 3000)
- At \( x = 1 \): \( y = 30,000 \cdot (3/4) = 22,500 \) (matches at \( x = 1\))
4. **Option d: \( y=30,000(4/3)^x \)**
- At \( x = 0 \): \( y = 30,000(1) = 30,000 \) (not 3000)
None of the options yield a perfect fit based on the initial value and the subsequent value. The one that nicely matches at \( x = 1 \) is option c, but not the initial point.
Given the analysis, and that none fit perfectly, we need to conclude:
The derived equation based on the points given would be closely represented in the form but does not exactly match any option, while option b does yield a correct 22500 but incorrect 3000. Therefore, based on growth factor and match:
**The best matching option is b: y=0.75(30,000)^x**, but it doesn't satisfy the initial value at x=0.
So we emphasize on (0,3000) and (1,22500) leads us primarily to construct new which ideally fits as \( y = 3000 (7.5)^x \).
We recommend re-evaluation of conditions/options based on consistent exponential operations and usage.
The general form of an exponential equation can be represented as:
\[ y = a \cdot b^x \]
Where:
- \( a \) is the initial value (the value at \( x = 0 \)).
- \( b \) is the growth (or decay) factor.
First, we identify the initial value from the point (0, 3000):
- At \( x = 0 \): \( y = 3000 \)
This means \( a = 3000 \).
Next, we can use the point (1, 22500) to find the value of \( b \):
Using the point (1, 22500):
\[ 22500 = 3000 \cdot b^1 \]
To solve for \( b \), we divide both sides by 3000:
\[ b = \frac{22500}{3000} = 7.5 \]
So, we can rewrite the equation as:
\[ y = 3000 \cdot 7.5^x \]
Now we need to check each of the given options and see if any match this relationship, particularly at \( x = 1 \):
1. **Option a: \( y=22,500(3/4)^x \)**
- At \( x = 1 \): \( y = 22,500(3/4) = 22,500 \cdot 0.75 = 16,875 \) (not 22500)
2. **Option b: \( y=0.75(30,000)^x \)**
- At \( x = 1 \): \( y = 0.75 \cdot 30,000 = 22,500 \) (this matches! But the equation at \( x = 0 \) does not match)
3. **Option c: \( y=30,000(3/4)^x \)**
- At \( x = 0 \): \( y = 30,000(1) = 30,000 \) (not 3000)
- At \( x = 1 \): \( y = 30,000 \cdot (3/4) = 22,500 \) (matches at \( x = 1\))
4. **Option d: \( y=30,000(4/3)^x \)**
- At \( x = 0 \): \( y = 30,000(1) = 30,000 \) (not 3000)
None of the options yield a perfect fit based on the initial value and the subsequent value. The one that nicely matches at \( x = 1 \) is option c, but not the initial point.
Given the analysis, and that none fit perfectly, we need to conclude:
The derived equation based on the points given would be closely represented in the form but does not exactly match any option, while option b does yield a correct 22500 but incorrect 3000. Therefore, based on growth factor and match:
**The best matching option is b: y=0.75(30,000)^x**, but it doesn't satisfy the initial value at x=0.
So we emphasize on (0,3000) and (1,22500) leads us primarily to construct new which ideally fits as \( y = 3000 (7.5)^x \).
We recommend re-evaluation of conditions/options based on consistent exponential operations and usage.