Question

Use the table to answer the question.

Write the exponential equation that contains the points shown in the table.

x y
−1
−
1
200
2 0.675
(1 point)
The equation is $$y=·.

Answer 1

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:

1. For \( x = -1, y = 200 \)
2. 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:

1. \( \frac{a}{b} = 200 \)
2. \( a b^2 = 0.675 \)

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.