Question
Use the equation of the polynomial function f(x)equals5 x cubed minus x to complete the following.
The curve opens
▼
to the right because the leading coefficient is
▼
negative
positive
. Because the polynomial is
▼
quartic,
cubic,
the graph has end behaviors in the
▼
same
opposite
direction, so the other end opens
▼
up
down
to the left.
The curve opens
▼
to the right because the leading coefficient is
▼
negative
positive
. Because the polynomial is
▼
quartic,
cubic,
the graph has end behaviors in the
▼
same
opposite
direction, so the other end opens
▼
up
down
to the left.
Answers
GPT-4o mini
To analyze the polynomial function \( f(x) = 5x^3 - x \), let's break down the statements given in the prompt.
1. **The curve opens**: Since the leading term is \( 5x^3 \), which has a positive coefficient, the curve opens to the right.
2. **Because the leading coefficient is**: The leading coefficient is positive.
3. **Because the polynomial is**: This polynomial is cubic (the highest degree of \( x \) is 3).
4. **The graph has end behaviors in the**: For a cubic function, the end behaviors are in opposite directions. Specifically, as \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity, and as \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity.
5. **So the other end opens**: Since one end approaches positive infinity and the other end approaches negative infinity, the other end opens down.
So, putting it all together, we have:
- The curve opens to the right because the leading coefficient is **positive**.
- Because the polynomial is **cubic**, the graph has end behaviors in the **opposite** direction, so the other end opens **down**.
1. **The curve opens**: Since the leading term is \( 5x^3 \), which has a positive coefficient, the curve opens to the right.
2. **Because the leading coefficient is**: The leading coefficient is positive.
3. **Because the polynomial is**: This polynomial is cubic (the highest degree of \( x \) is 3).
4. **The graph has end behaviors in the**: For a cubic function, the end behaviors are in opposite directions. Specifically, as \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity, and as \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity.
5. **So the other end opens**: Since one end approaches positive infinity and the other end approaches negative infinity, the other end opens down.
So, putting it all together, we have:
- The curve opens to the right because the leading coefficient is **positive**.
- Because the polynomial is **cubic**, the graph has end behaviors in the **opposite** direction, so the other end opens **down**.