Questions: Use end behavior and analysis of the dominating term to determine which function matches the graph. Which function matches the graph? A. f(x)=-2x^4+x^3-4x^2+3x+3 B. f(x)=x^5-4x^2+3x+3 C. f(x)=-x^5-4x^2+3x+3 D. f(x)=2x^4+x^3-4x^2+3x+3

Use end behavior and analysis of the dominating term to determine which function matches the graph. Which function matches the graph? A. f(x)=-2x^4+x^3-4x^2+3x+3 B. f(x)=x^5-4x^2+3x+3 C. f(x)=-x^5-4x^2+3x+3 D. f(x)=2x^4+x^3-4x^2+3x+3
Transcript text: Use end behavior and analysis of the dominating term to determine which function matches the graph. Which function matches the graph? A. $f(x)=-2x^4+x^3-4x^2+3x+3$ B. $f(x)=x^5-4x^2+3x+3$ C. $f(x)=-x^5-4x^2+3x+3$ D. $f(x)=2x^4+x^3-4x^2+3x+3$
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Solution

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Solution Steps

Step 1: Identify the Dominant Term

The dominant term in a polynomial function is the term with the highest power of \( x \). This term will determine the end behavior of the function.

Step 2: Analyze the End Behavior

Examine the graph to determine the end behavior. For example, if the graph rises to the left and falls to the right, the dominant term will have a negative coefficient and an odd power.

Step 3: Match the Dominant Term with the Options

Compare the end behavior of the graph with the dominant terms of the given options:

  • Option A: \( x^4 \)
  • Option B: \( -x^4 \)
  • Option C: \( -x^3 \)
  • Option D: \( x^3 \)

Final Answer

Based on the end behavior observed in the graph, the function that matches the graph is: \[ f(x) = -x^3 + 2x^2 - x + 3 \] (Option C)

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