Question

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Explanation:

Analyze key features of the function

The given function is:

$$ y = -(2)^x - 1 $$

We identify the following characteristics:

1. Asymptote: As \(x \to -\infty\), \(2^x \to 0\), so \(y \to -1\). The horizontal asymptote is \(y = -1\).
2. y-intercept: Setting \(x = 0\):

$$ y = -(2)^0 - 1 = -1 - 1 = -2 $$

The y-intercept is at \((0, -2)\).

1. Other points:

• For \(x = 1\), \(y = -(2)^1 - 1 = -3\).
• For \(x = 2\), \(y = -(2)^2 - 1 = -5\).
• For \(x = 3\), \(y = -(2)^3 - 1 = -9\).

Match with the visible graph

Looking at the first graph option shown in the image:

• The curve approaches a horizontal line near \(y = -1\) as \(x\) goes to the left (negative x-direction).
• The curve passes through the y-intercept at \((0, -2)\).
• The curve passes through \((1, -3)\), \((2, -5)\), and \((3, -9)\).

This perfectly matches the mathematical features of the function \(y = -(2)^x - 1\).

Answer:

• (A) The first graph, which has a horizontal asymptote at \(y = -1\), a y-intercept at \((0, -2)\), and passes through \((1, -3)\), \((2, -5)\), and \((3, -9)\). (Correct answer)