To find the exponential equation that fits the points shown in the table, we need to assume the form of the exponential function, which is typically given as:
\[ y = ab^x \]
Where:
- \( a \) is the initial value (when \( x = 0 \)),
- \( b \) is the base (growth or decay factor).
From the provided points:
- For \( x = -1, y = 200 \)
- For \( x = 2, y = 0.675 \)
We can use these points to create two equations.
Step 1: Set Up the Equations
From the first point \( (-1, 200) \):
\[ 200 = ab^{-1} \]
This can be rewritten as:
\[ ab^{-1} = 200 \quad \Rightarrow \quad \frac{a}{b} = 200 \quad \text{(Equation 1)} \]
From the second point \( (2, 0.675) \):
\[ 0.675 = ab^2 \quad \text{(Equation 2)} \]
Step 2: Solve the Equations
Now we have a system of equations:
- \( \frac{a}{b} = 200 \)
- \( a b^2 = 0.675 \)
From Equation 1, we can express \( a \) in terms of \( b \):
\[ a = 200b \]
Now, substitute \( a \) into Equation 2:
\[ 200b \cdot b^2 = 0.675 \]
This simplifies to:
\[ 200b^3 = 0.675 \]
Now, solve for \( b \):
\[ b^3 = \frac{0.675}{200} = 0.003375 \]
Taking the cube root of both sides:
\[ b = \sqrt[3]{0.003375} \approx 0.15 \]
Step 3: Find \( a \)
Now substitute \( b \) back into the equation for \( a \):
\[ a = 200b \approx 200 \cdot 0.15 = 30 \]
Final Equation
Thus, the exponential equation becomes:
\[ y = 30(0.15)^x \]
So, we can write the equation as:
\[ \boxed{y = 30(0.15)^x} \]
This is the exponential equation that passes through the given points.
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