Question
Use the equation of the polynomial function f(x)equalsleft parenthesis x minus 1 right parenthesis left parenthesis x squared minus 4 right parenthesis to complete the following.
(a) State the degree and the leading coefficient.
(b) Describe the end behavior of the graph of the function.
(c) Support your answer by graphing the function.
Question content area bottom
Part 1
(a) The degree of the polynomial is
enter your response here and the leading coefficient is
enter your response here.
(a) State the degree and the leading coefficient.
(b) Describe the end behavior of the graph of the function.
(c) Support your answer by graphing the function.
Question content area bottom
Part 1
(a) The degree of the polynomial is
enter your response here and the leading coefficient is
enter your response here.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To analyze the polynomial function \( f(x) = (x - 1)(x^2 - 4) \), we'll first start with the necessary calculations.
### Part 1: Degree and Leading Coefficient
(a) **Finding the degree and leading coefficient:**
1. The polynomial can be rewritten as:
\[
f(x) = (x - 1)(x^2 - 4)
\]
The term \( (x^2 - 4) \) can be factored further, but here we can still analyze it in its product form.
2. The degree of each polynomial factor is:
- \( (x - 1) \) has a degree of 1.
- \( (x^2 - 4) \) has a degree of 2.
3. To find the overall degree of the polynomial, we add the degrees of the factors:
\[
\text{Degree of } f(x) = 1 + 2 = 3.
\]
4. The leading coefficient comes from multiplying the leading coefficients of each factor. The leading coefficient of \( (x - 1) \) is 1, and the leading coefficient of \( (x^2 - 4) \) is also 1. Therefore, the leading coefficient of \( f(x) \) is:
\[
1 \cdot 1 = 1.
\]
Thus, **the degree of the polynomial is 3**, and **the leading coefficient is 1**.
### Part 2: End Behavior
(b) **Describing the end behavior of the graph of the function:**
- Since the degree of the polynomial is odd (3) and the leading coefficient is positive (1), we can describe the end behavior as follows:
- As \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \).
### Part 3: Graphing the Function
(c) To graph the function, we can find critical points, such as the roots:
1. **Roots of the function:** Set \( f(x) = 0 \):
\[
f(x) = (x - 1)(x^2 - 4) = 0.
\]
This gives us:
- \( x - 1 = 0 \) ⇒ \( x = 1 \)
- \( x^2 - 4 = 0 \) ⇒ \( x^2 = 4 \) ⇒ \( x = 2 \) or \( x = -2 \)
Therefore, the roots are \( x = 1, 2, -2 \).
2. **Sketching the graph:**
- Mark the points where the function crosses the x-axis: \( (-2, 0), (1, 0), (2, 0) \).
- Since the leading coefficient is positive, the graph will rise to the right and fall to the left.
- Analyze typical points or the y-intercept to get a better graph shape (e.g., evaluate \( f(0) \)):
\[
f(0) = (0 - 1)(0^2 - 4) = (-1)(-4) = 4.
\]
So, the y-intercept is (0, 4).
Using this information, you can sketch the function, ensuring it crosses the x-axis at the identified points and reflects the described end behavior.
In conclusion:
- **(a)** Degree: **3**, Leading coefficient: **1**.
- **(b)** End behavior: Rises to \( +\infty \) as \( x \to +\infty \); falls to \( -\infty \) as \( x \to -\infty \).
### Part 1: Degree and Leading Coefficient
(a) **Finding the degree and leading coefficient:**
1. The polynomial can be rewritten as:
\[
f(x) = (x - 1)(x^2 - 4)
\]
The term \( (x^2 - 4) \) can be factored further, but here we can still analyze it in its product form.
2. The degree of each polynomial factor is:
- \( (x - 1) \) has a degree of 1.
- \( (x^2 - 4) \) has a degree of 2.
3. To find the overall degree of the polynomial, we add the degrees of the factors:
\[
\text{Degree of } f(x) = 1 + 2 = 3.
\]
4. The leading coefficient comes from multiplying the leading coefficients of each factor. The leading coefficient of \( (x - 1) \) is 1, and the leading coefficient of \( (x^2 - 4) \) is also 1. Therefore, the leading coefficient of \( f(x) \) is:
\[
1 \cdot 1 = 1.
\]
Thus, **the degree of the polynomial is 3**, and **the leading coefficient is 1**.
### Part 2: End Behavior
(b) **Describing the end behavior of the graph of the function:**
- Since the degree of the polynomial is odd (3) and the leading coefficient is positive (1), we can describe the end behavior as follows:
- As \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \).
### Part 3: Graphing the Function
(c) To graph the function, we can find critical points, such as the roots:
1. **Roots of the function:** Set \( f(x) = 0 \):
\[
f(x) = (x - 1)(x^2 - 4) = 0.
\]
This gives us:
- \( x - 1 = 0 \) ⇒ \( x = 1 \)
- \( x^2 - 4 = 0 \) ⇒ \( x^2 = 4 \) ⇒ \( x = 2 \) or \( x = -2 \)
Therefore, the roots are \( x = 1, 2, -2 \).
2. **Sketching the graph:**
- Mark the points where the function crosses the x-axis: \( (-2, 0), (1, 0), (2, 0) \).
- Since the leading coefficient is positive, the graph will rise to the right and fall to the left.
- Analyze typical points or the y-intercept to get a better graph shape (e.g., evaluate \( f(0) \)):
\[
f(0) = (0 - 1)(0^2 - 4) = (-1)(-4) = 4.
\]
So, the y-intercept is (0, 4).
Using this information, you can sketch the function, ensuring it crosses the x-axis at the identified points and reflects the described end behavior.
In conclusion:
- **(a)** Degree: **3**, Leading coefficient: **1**.
- **(b)** End behavior: Rises to \( +\infty \) as \( x \to +\infty \); falls to \( -\infty \) as \( x \to -\infty \).
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