Question

which is the graph of \\(f(x) = 4^x\\)?

Explanation:

Analyze the function type

Using the Exponential Function Graphs and Exponential Growth and Decay Behavior knowledge points

$$ f(x) = 4^x $$

This is an exponential growth function because the base \(b = 4\) is greater than \(1\).

Determine key graph characteristics

Using the Exponential Function Graphs knowledge point

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The graph must have a \(y\)-intercept at \((0, 1)\), pass through \((1, 4)\), and have a horizontal asymptote at \(y = 0\) as \(x \to -\infty\).

Compare with the given options

Using the Exponential Function Graphs knowledge point

• The first graph shows a curve starting near the negative \(x\)-axis, passing through \((0, 1)\), and rising rapidly to the right. This matches the exponential growth function \(f(x) = 4^x\).
• The second graph is a U-shaped curve (resembling a polynomial of even degree, like a flat-bottomed quartic or similar), which does not represent an exponential function.

Answer:

• (A) The first graph (exponential curve passing through \((0,1)\)) (Correct answer)
• (B) The second graph (U-shaped curve)