Question
For the polynomial function f(x)equalsnegative 10 x cubed plus 25 x plus x Superscript 5, answer the parts a through e.
Question content area bottom
Part 1
a. Use the Leading Coefficient Test to determine the graph's end behavior.
A.
The graph of f(x) rises to the left and falls to the right.
B.
The graph of f(x) falls to the left and falls to the right.
C.
The graph of f(x) falls to the left and rises to the right.
D.
The graph of f(x) rises to the left and rises to the right.
Question content area bottom
Part 1
a. Use the Leading Coefficient Test to determine the graph's end behavior.
A.
The graph of f(x) rises to the left and falls to the right.
B.
The graph of f(x) falls to the left and falls to the right.
C.
The graph of f(x) falls to the left and rises to the right.
D.
The graph of f(x) rises to the left and rises to the right.
Answers
Answered by
GPT-4o mini
To analyze the end behavior of the polynomial function \( f(x) = -10x^3 + 25x + x^5 \), we need to consider the leading term, which is the term with the highest degree.
In this case, the leading term is \( x^5 \).
### Step 1: Identify the leading term
- The leading term is \( x^5 \), which has an odd degree (5).
### Step 2: Identify the leading coefficient
- The leading coefficient for \( x^5 \) is \( 1 \), which is positive.
### Step 3: Apply the Leading Coefficient Test
- For polynomials, the end behavior is determined as follows:
- If the leading term has an **odd degree** and a **positive coefficient**, the graph will **rise to the right** and **fall to the left**.
- If the leading term has an **odd degree** and a **negative coefficient**, the graph will **fall to the right** and **rise to the left**.
- If the leading term has an **even degree** and a **positive coefficient**, the graph will **rise on both sides**.
- If the leading term has an **even degree** and a **negative coefficient**, the graph will **fall on both sides**.
### Conclusion
In our case:
- Since \( x^5 \) is an odd degree with a positive leading coefficient, the graph of \( f(x) \) falls to the left and rises to the right.
Thus, the answer is:
**C. The graph of f(x) falls to the left and rises to the right.**
In this case, the leading term is \( x^5 \).
### Step 1: Identify the leading term
- The leading term is \( x^5 \), which has an odd degree (5).
### Step 2: Identify the leading coefficient
- The leading coefficient for \( x^5 \) is \( 1 \), which is positive.
### Step 3: Apply the Leading Coefficient Test
- For polynomials, the end behavior is determined as follows:
- If the leading term has an **odd degree** and a **positive coefficient**, the graph will **rise to the right** and **fall to the left**.
- If the leading term has an **odd degree** and a **negative coefficient**, the graph will **fall to the right** and **rise to the left**.
- If the leading term has an **even degree** and a **positive coefficient**, the graph will **rise on both sides**.
- If the leading term has an **even degree** and a **negative coefficient**, the graph will **fall on both sides**.
### Conclusion
In our case:
- Since \( x^5 \) is an odd degree with a positive leading coefficient, the graph of \( f(x) \) falls to the left and rises to the right.
Thus, the answer is:
**C. The graph of f(x) falls to the left and rises to the right.**