Question
Given f(x) = (x − 1)(x + 2)(x − 3), what are the zeros and end behavior of the function?
−1, 2, −3; continues downward to the left and upward to the right
−1, 2, −3; continues upward to the left and downward to the right
1, −2, 3; continues downward to the left and upward to the right
1, −2, 3; continues upward to the left and downward to the right
−1, 2, −3; continues downward to the left and upward to the right
−1, 2, −3; continues upward to the left and downward to the right
1, −2, 3; continues downward to the left and upward to the right
1, −2, 3; continues upward to the left and downward to the right
Answers
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Answered by
GPT-4o mini
To find the zeros of the function \( f(x) = (x - 1)(x + 2)(x - 3) \), we set the function equal to zero and solve for \( x \):
1. \( x - 1 = 0 \) ⇒ \( x = 1 \)
2. \( x + 2 = 0 \) ⇒ \( x = -2 \)
3. \( x - 3 = 0 \) ⇒ \( x = 3 \)
Thus, the zeros of the function \( f(x) \) are \( x = 1, -2, 3 \).
Next, we analyze the end behavior of the function. The polynomial is of degree 3 (since the highest power of \( x \) in the expanded version is \( x^3 \)). The leading coefficient is positive, as it comes from the expansion of the function.
For a polynomial of odd degree with a positive leading coefficient:
- As \( x \to -\infty \), \( f(x) \to -\infty \) (the function goes downward to the left).
- As \( x \to +\infty \), \( f(x) \to +\infty \) (the function goes upward to the right).
Combining this information, we have:
- Zeros: \( 1, -2, 3 \)
- End behavior: continues downward to the left and upward to the right.
The correct choice that matches this analysis is:
**1, -2, 3; continues downward to the left and upward to the right.**
1. \( x - 1 = 0 \) ⇒ \( x = 1 \)
2. \( x + 2 = 0 \) ⇒ \( x = -2 \)
3. \( x - 3 = 0 \) ⇒ \( x = 3 \)
Thus, the zeros of the function \( f(x) \) are \( x = 1, -2, 3 \).
Next, we analyze the end behavior of the function. The polynomial is of degree 3 (since the highest power of \( x \) in the expanded version is \( x^3 \)). The leading coefficient is positive, as it comes from the expansion of the function.
For a polynomial of odd degree with a positive leading coefficient:
- As \( x \to -\infty \), \( f(x) \to -\infty \) (the function goes downward to the left).
- As \( x \to +\infty \), \( f(x) \to +\infty \) (the function goes upward to the right).
Combining this information, we have:
- Zeros: \( 1, -2, 3 \)
- End behavior: continues downward to the left and upward to the right.
The correct choice that matches this analysis is:
**1, -2, 3; continues downward to the left and upward to the right.**
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