Questions: Which of the following graphs represents the function f(x) = x^4 - 2x^3 - 3x^2 + 4x + 1?

Which of the following graphs represents the function f(x) = x^4 - 2x^3 - 3x^2 + 4x + 1?
Transcript text: Which of the following graphs represents the function $f(x)=x^{4}-2 x^{3}-$ $3 x^{2}+4 x+1 ?$
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Solution

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Solution Steps

Step 1: Identify the function and its characteristics

The given function is \( f(x) = x^4 - 2x^3 - 3x^2 + 4x + 1 \). This is a polynomial of degree 4, which means its graph will have at most 3 turning points and will generally have a "W" shape.

Step 2: Analyze the end behavior

For large positive or negative values of \( x \), the \( x^4 \) term will dominate. Since the coefficient of \( x^4 \) is positive, the graph will rise to positive infinity as \( x \) approaches both positive and negative infinity.

Step 3: Determine the number of turning points

A polynomial of degree 4 can have up to 3 turning points. We need to check which graph has this characteristic.

Step 4: Compare the graphs
  • Graph 1 has 3 turning points and the end behavior matches the expected behavior (rising to positive infinity on both ends).
  • Graph 2 has 2 turning points and does not match the expected end behavior for a degree 4 polynomial.

Final Answer

The graph that represents the function \( f(x) = x^4 - 2x^3 - 3x^2 + 4x + 1 \) is:

Graph 1

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