Which of the following scatterplot could be modeled by the equation `y=(a)/(b^(x))`, where a and b are constant such that `agt1 and xlt-1`?

Options

A.
B.
C.
D.

Explanation

To solve the problem of identifying which scatterplot could be modeled by the equation \( y = \frac{a}{b^x} \) where \( a > 1 \) and \( x < -1 \), we can follow these steps:

### Step 1: Understand the Equation
The equation \( y = \frac{a}{b^x} \) can be rewritten as \( y = a \cdot b^{-x} \). This shows that as \( x \) increases (becomes less negative), \( y \) will decrease because \( b^{-x} \) will increase.

**Hint:** Rewrite the equation in a form that makes it easier to analyze the behavior of \( y \) as \( x \) changes.

### Step 2: Analyze the Conditions
Given that \( a > 1 \) and \( x < -1 \), we know:
- \( a \) is a positive constant greater than 1.
- Since \( x < -1 \), the values of \( x \) are negative, which means \( -x \) is positive.

**Hint:** Consider how the value of \( x \) affects the output \( y \) based on the conditions provided.

### Step 3: Determine the Behavior of the Function
As \( x \) becomes more negative (i.e., \( x \) approaches negative infinity), \( b^{-x} \) will grow larger. Thus, \( y \) will increase without bound. Conversely, as \( x \) approaches -1 from the left, \( y \) will be at a finite value.

**Hint:** Think about how the graph of the function behaves as \( x \) changes, especially for values less than -1.

### Step 4: Evaluate the Scatterplots
Now, we need to look at the provided scatterplots (let's assume they are labeled A, B, C, D) and determine which one reflects the behavior we analyzed:
- The correct scatterplot should show an increasing trend for \( x < -1 \).

**Hint:** Look for a scatterplot that shows an upward trend as \( x \) values become less negative.

### Step 5: Eliminate Incorrect Options
From the analysis, we can eliminate options that do not show the expected behavior:
- If a scatterplot shows a decreasing trend or is flat for \( x < -1 \), it cannot be modeled by our equation.

**Hint:** Cross out any scatterplots that do not match the increasing behavior for negative \( x \).

### Step 6: Identify the Correct Scatterplot
After evaluating the options, we find that scatterplot B shows the correct increasing trend for \( x < -1 \), while scatterplots A, C, and D do not fit the model.

**Final Answer:** The correct scatterplot that could be modeled by the equation \( y = \frac{a}{b^x} \) is **B**.