Question
I got a graph with a point at one, 14 and another point at 2:00, 4 given the graph of the exponential equation which option is the correct exponential equation for the graph
Answers
Answered by
GPT-4o mini
To determine the correct exponential equation for the graph given two points, we first need to identify the points on the graph clearly. The points you provided are:
1. \( (1, 14) \)
2. \( (2, 4) \)
The general form of an exponential function is:
\[
y = ab^x
\]
where \( a \) is the initial value (the value when \( x = 0 \)), and \( b \) is the base of the exponential function.
Using the two points to set up a system of equations:
1. For the point \( (1, 14) \):
\[
14 = ab^1 \quad \text{(1)}
\]
2. For the point \( (2, 4) \):
\[
4 = ab^2 \quad \text{(2)}
\]
Now let's solve these equations.
From equation (1):
\[
ab = 14 \quad \text{(3)}
\]
From equation (2):
\[
ab^2 = 4 \quad \text{(4)}
\]
We can substitute equation (3) into equation (4) to find \( b \):
Substituting \( a = \frac{14}{b} \) into equation (4):
\[
\frac{14}{b} b^2 = 4
\]
This simplifies to:
\[
14b = 4
\]
\[
b = \frac{4}{14} = \frac{2}{7}
\]
Now let's substitute \( b \) back into equation (3) to find \( a \):
\[
a \left(\frac{2}{7}\right) = 14
\]
\[
a = 14 \cdot \frac{7}{2} = 49
\]
Now we have \( a \) and \( b \):
\[
a = 49, \quad b = \frac{2}{7}
\]
Thus, the exponential function becomes:
\[
y = 49\left(\frac{2}{7}\right)^x
\]
You can simplify this slightly but this form demonstrates the essential characteristics of the function based on the points you've given. If you're provided with multiple choice options, you should look for an option that matches \( y = 49\left(\frac{2}{7}\right)^x \) or a similar representation.
1. \( (1, 14) \)
2. \( (2, 4) \)
The general form of an exponential function is:
\[
y = ab^x
\]
where \( a \) is the initial value (the value when \( x = 0 \)), and \( b \) is the base of the exponential function.
Using the two points to set up a system of equations:
1. For the point \( (1, 14) \):
\[
14 = ab^1 \quad \text{(1)}
\]
2. For the point \( (2, 4) \):
\[
4 = ab^2 \quad \text{(2)}
\]
Now let's solve these equations.
From equation (1):
\[
ab = 14 \quad \text{(3)}
\]
From equation (2):
\[
ab^2 = 4 \quad \text{(4)}
\]
We can substitute equation (3) into equation (4) to find \( b \):
Substituting \( a = \frac{14}{b} \) into equation (4):
\[
\frac{14}{b} b^2 = 4
\]
This simplifies to:
\[
14b = 4
\]
\[
b = \frac{4}{14} = \frac{2}{7}
\]
Now let's substitute \( b \) back into equation (3) to find \( a \):
\[
a \left(\frac{2}{7}\right) = 14
\]
\[
a = 14 \cdot \frac{7}{2} = 49
\]
Now we have \( a \) and \( b \):
\[
a = 49, \quad b = \frac{2}{7}
\]
Thus, the exponential function becomes:
\[
y = 49\left(\frac{2}{7}\right)^x
\]
You can simplify this slightly but this form demonstrates the essential characteristics of the function based on the points you've given. If you're provided with multiple choice options, you should look for an option that matches \( y = 49\left(\frac{2}{7}\right)^x \) or a similar representation.