Question

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Explanation:

Identify the base function and its features

The given function is \(f(x) = 2^x\).

• Domain: \((-\infty, \infty)\)
• Range: \((0, \infty)\)
• Horizontal Asymptote: \(y = 0\)
• \(y\)-intercept: \((0, 1)\)
• Key point: \((1, 2)\)

Analyze standard transformations and their corresponding features

Since the matching options are cut off, we determine the standard transformations of \(f(x) = 2^x\) and their key features:

• Vertical translation \(g(x) = 2^x + k\):
• Shifts the graph vertically by \(k\) units.
• New horizontal asymptote: \(y = k\).
• New \(y\)-intercept: \((0, 1 + k)\).
• Horizontal translation \(g(x) = 2^{x - h}\):
• Shifts the graph horizontally by \(h\) units.
• Horizontal asymptote remains \(y = 0\).
• New \(y\)-intercept: \((0, 2^{-h})\).
• Vertical stretch/compression \(g(x) = a \cdot 2^x\):
• New \(y\)-intercept: \((0, a)\).
• Horizontal asymptote remains \(y = 0\).
• Reflection across the x-axis \(g(x) = -2^x\):
• New horizontal asymptote: \(y = 0\).
• New \(y\)-intercept: \((0, -1)\).
• Range becomes \((-\infty, 0)\).
• Reflection across the y-axis \(g(x) = 2^{-x}\):
• New horizontal asymptote: \(y = 0\).
• New \(y\)-intercept: \((0, 1)\).
• Decreasing function instead of increasing.

Formulate the general matching rules

• Any transformation of the form \(f(x) + k\) shifts the horizontal asymptote to \(y = k\).
• Any transformation of the form \(a \cdot f(x)\) changes the \(y\)-intercept to \((0, a)\).
• Any transformation of the form \(f(x - h)\) shifts key points horizontally by \(h\) units.

Answer:

Based on the base function \(f(x) = 2^x\), the transformations match to their features as follows:

1. Vertical Shift \(g(x) = 2^x + k\):

• Feature: The horizontal asymptote shifts to \(y = k\), and the \(y\)-intercept becomes \((0, 1 + k)\).

1. Horizontal Shift \(g(x) = 2^{x - h}\):

• Feature: The graph shifts horizontally, changing the \(y\)-intercept to \((0, 2^{-h})\) while keeping the horizontal asymptote at \(y = 0\).

1. Vertical Stretch/Compression \(g(x) = a \cdot 2^x\):

• Feature: The \(y\)-intercept changes to \((0, a)\) while keeping the horizontal asymptote at \(y = 0\).

1. Reflection across the x-axis \(g(x) = -2^x\):

• Feature: The graph is reflected vertically, changing the \(y\)-intercept to \((0, -1)\) and the range to \((-\infty, 0)\).

1. Reflection across the y-axis \(g(x) = 2^{-x}\):

• Feature: The graph is reflected horizontally, changing the function from increasing to decreasing while keeping the \(y\)-intercept at \((0, 1)\).