Question

which functions represent exponential growth?
\\(y = f(x)\\)
\\(y = h(x)\\)
\\(y = g(x)\\)
\\(y = k(x)\\)

which functions have a horizontal asymptote?
\\(y = f(x)\\)
\\(y = h(x)\\)
\\(y = g(x)\\)
\\(y = k(x)\\)

Explanation:

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<reasoning>

Identify exponential growth functions

Using the Exponential Growth and Decay Behavior knowledge point

• The functions \(y = h(x)\) (green curve) and \(y = k(x)\) (purple curve) increase from left to right, approaching \(0\) as \(x \to -\infty\) and growing rapidly as \(x \to \infty\). Thus, \(y = h(x)\) and \(y = k(x)\) represent exponential growth.

Identify horizontal asymptotes

Using the Horizontal Asymptotes knowledge point

• An exponential function of the form \(y = ab^x\) (where \(a > 0, b > 0\)) has a horizontal asymptote at \(y = 0\) (the x-axis).
• Looking at the graph, all four curves—\(y = f(x)\) (orange), \(y = g(x)\) (blue), \(y = h(x)\) (green), and \(y = k(x)\) (purple)—flatten out and approach the x-axis (\(y = 0\)) on one side.
• Specifically, \(y = f(x)\) and \(y = g(x)\) approach \(y = 0\) as \(x \to \infty\).
• Meanwhile, \(y = h(x)\) and \(y = k(x)\) approach \(y = 0\) as \(x \to -\infty\).
• Therefore, all four functions have a horizontal asymptote.

</reasoning>

<answer>
<mcq-correct>[x] \(y = f(x)\)</mcq-correct>
<mcq-correct>[x] \(y = h(x)\)</mcq-correct>
<mcq-correct>[x] \(y = g(x)\)</mcq-correct>
<mcq-correct>[x] \(y = k(x)\)</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
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"Exponential Function Graphs"
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Answer:

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<reasoning>

Identify exponential growth functions

Using the Exponential Growth and Decay Behavior knowledge point

• The functions \(y = h(x)\) (green curve) and \(y = k(x)\) (purple curve) increase from left to right, approaching \(0\) as \(x \to -\infty\) and growing rapidly as \(x \to \infty\). Thus, \(y = h(x)\) and \(y = k(x)\) represent exponential growth.

Identify horizontal asymptotes

Using the Horizontal Asymptotes knowledge point

• An exponential function of the form \(y = ab^x\) (where \(a > 0, b > 0\)) has a horizontal asymptote at \(y = 0\) (the x-axis).
• Looking at the graph, all four curves—\(y = f(x)\) (orange), \(y = g(x)\) (blue), \(y = h(x)\) (green), and \(y = k(x)\) (purple)—flatten out and approach the x-axis (\(y = 0\)) on one side.
• Specifically, \(y = f(x)\) and \(y = g(x)\) approach \(y = 0\) as \(x \to \infty\).
• Meanwhile, \(y = h(x)\) and \(y = k(x)\) approach \(y = 0\) as \(x \to -\infty\).
• Therefore, all four functions have a horizontal asymptote.

</reasoning>

<answer>
<mcq-correct>[x] \(y = f(x)\)</mcq-correct>
<mcq-correct>[x] \(y = h(x)\)</mcq-correct>
<mcq-correct>[x] \(y = g(x)\)</mcq-correct>
<mcq-correct>[x] \(y = k(x)\)</mcq-correct>
</answer>

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