Question

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Explanation:

Identify the parent function and its key features

The parent function is:

$$f(x) = \log_2 x$$

Key features of \(f(x)\):

• Vertical asymptote: \(x = 0\)
• Domain: \((0, \infty)\)
• Range: \((-\infty, \infty)\)
• Key point (\(x\)-intercept): \((1, 0)\)
• Another key point: \((2, 1)\)

Apply the transformations to find the features of \(g(x)\)

The transformed function is:

$$g(x) = f(x + 4) + 8 = \log_2(x + 4) + 8$$

The transformations applied to \(f(x)\) are:

• Horizontal shift: Left by \(4\) units (\(x \to x + 4\))
• Vertical shift: Up by \(8\) units (\(y \to y + 8\))

Determine the new features of \(g(x)\)

Applying these shifts to the key features of \(f(x)\):

• New vertical asymptote: \(x = 0 - 4 \implies x = -4\)
• New domain: \((-4, \infty)\)
• New range: \((-\infty, \infty)\)
• Transformed point \((1, 0)\): \((1 - 4, 0 + 8) = (-3, 8)\)
• Transformed point \((2, 1)\): \((2 - 4, 1 + 8) = (-2, 9)\)

Answer:

The features of the function \(g(x) = \log_2(x + 4) + 8\) are:

• Vertical Asymptote: \(x = -4\)
• Domain: \((-4, \infty)\) or \(x > -4\)
• Range: \((-\infty, \infty)\) or all real numbers
• Key Points on the Graph: \((-3, 8)\) and \((-2, 9)\)