Question
Given the graph of the exponential equation that goes through the points (0, 3) and (1, 4), which of the following is the correct exponential equation for the graph?
(1 point)
Responses
y=3(4/3)^x
y is equal to 3 times 4 thirds to the x th power
y=3(3/4)^x
y is equal to 3 times 3 fourths to the x th power
y=3(1/4)^x
y is equal to 3 times 1 fourth to the x th power
y=4(1/3)^x
(1 point)
Responses
y=3(4/3)^x
y is equal to 3 times 4 thirds to the x th power
y=3(3/4)^x
y is equal to 3 times 3 fourths to the x th power
y=3(1/4)^x
y is equal to 3 times 1 fourth to the x th power
y=4(1/3)^x
Answers
Answered by
GPT-4o mini
To find the correct exponential equation, we can start with the general form of an exponential function, which is:
\[ y = a \cdot b^x \]
where \( a \) is the initial value (the y-intercept), and \( b \) is the base. The point \( (0, 3) \) gives us \( a = 3 \).
This means we can rewrite our equation as:
\[ y = 3 \cdot b^x \]
Next, we use the second point \( (1, 4) \) to find \( b \):
When \( x = 1 \), \( y = 4 \):
\[ 4 = 3 \cdot b^1 \]
\[ 4 = 3b \]
Now, we can solve for \( b \):
\[ b = \frac{4}{3} \]
Thus, we can write the complete equation as:
\[ y = 3 \cdot \left(\frac{4}{3}\right)^x \]
Checking through the options provided, we see that the correct equation is:
**y = 3(4/3)^x**
\[ y = a \cdot b^x \]
where \( a \) is the initial value (the y-intercept), and \( b \) is the base. The point \( (0, 3) \) gives us \( a = 3 \).
This means we can rewrite our equation as:
\[ y = 3 \cdot b^x \]
Next, we use the second point \( (1, 4) \) to find \( b \):
When \( x = 1 \), \( y = 4 \):
\[ 4 = 3 \cdot b^1 \]
\[ 4 = 3b \]
Now, we can solve for \( b \):
\[ b = \frac{4}{3} \]
Thus, we can write the complete equation as:
\[ y = 3 \cdot \left(\frac{4}{3}\right)^x \]
Checking through the options provided, we see that the correct equation is:
**y = 3(4/3)^x**