To predict the end behavior of the function \( f(x) = 3(x - 3)(x^2 + 3) \), we should analyze the degree of the polynomial and the leading term.
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Identify the degrees of each factor:
- The term \( (x - 3) \) is a linear polynomial of degree 1.
- The term \( (x^2 + 3) \) is a quadratic polynomial of degree 2.
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Calculate the total degree of the polynomial:
- The total degree of the function \( f(x) \) is \( 1 + 2 = 3 \).
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Determine the leading term:
- The leading term of the polynomial comes from the product of the leading terms of each factor. The leading term of \( (x - 3) \) is \( x \) and the leading term of \( (x^2 + 3) \) is \( x^2 \), so the leading term of \( f(x) \) is: \[ 3 \cdot x \cdot x^2 = 3x^3 \]
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Analyze the leading term for end behavior:
- Since the degree of the polynomial is odd (3) and the leading coefficient (3) is positive, we can predict the end behavior:
- As \( x \to -\infty \) (approaching negative infinity), \( f(x) \to -\infty \).
- As \( x \to \infty \) (approaching positive infinity), \( f(x) \to \infty \).
- Since the degree of the polynomial is odd (3) and the leading coefficient (3) is positive, we can predict the end behavior:
In summary, the end behavior of the graph of the function \( f(x) = 3(x - 3)(x^2 + 3) \) is:
- \( f(x) \to -\infty \) as \( x \to -\infty \)
- \( f(x) \to \infty \) as \( x \to \infty \)
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