Asked by blobby
Use the image to answer the question.
A curve is graphed on a coordinate plane. The plane’s x-axis ranges from negative 5 to 5 in unit increments but labeled every 5 units. The y-axis ranges from negative 10 to 10 in unit increments but labeled at every 5 units.
The equation y=x4+3x3−x2−3x
has four solutions and is shown in the graph. Which of the following student statements is true?
Statement #1: The equation has four non-real solutions because there are four x-intercepts.
Statement #2: The equation has three real solutions because there are three turning points on the graph.
Statement #3: The equation has four real solutions because there are four x-intercepts.
(1 point)
Statement #
is true.
A curve is graphed on a coordinate plane. The plane’s x-axis ranges from negative 5 to 5 in unit increments but labeled every 5 units. The y-axis ranges from negative 10 to 10 in unit increments but labeled at every 5 units.
The equation y=x4+3x3−x2−3x
has four solutions and is shown in the graph. Which of the following student statements is true?
Statement #1: The equation has four non-real solutions because there are four x-intercepts.
Statement #2: The equation has three real solutions because there are three turning points on the graph.
Statement #3: The equation has four real solutions because there are four x-intercepts.
(1 point)
Statement #
is true.
Answers
Answered by
GPT-4o mini
To determine which statement is true, we need to analyze the equation and its graphical representation.
### Analyzing the Statements
1. **Statement #1**: "The equation has four non-real solutions because there are four x-intercepts."
- This statement is false. If there are four x-intercepts, it means the equation has four real solutions since x-intercepts correspond to the solutions of the equation \( y = 0 \).
2. **Statement #2**: "The equation has three real solutions because there are three turning points on the graph."
- This statement is also false. The number of turning points does not directly correlate with the number of solutions. A polynomial can have more turning points than solutions.
3. **Statement #3**: "The equation has four real solutions because there are four x-intercepts."
- This statement is true. The equation being a polynomial of degree 4 can have up to 4 real solutions, and having four x-intercepts indicates that there are indeed four real solutions.
### Conclusion
Thus, the correct statement is:
**Statement #3 is true.**
### Analyzing the Statements
1. **Statement #1**: "The equation has four non-real solutions because there are four x-intercepts."
- This statement is false. If there are four x-intercepts, it means the equation has four real solutions since x-intercepts correspond to the solutions of the equation \( y = 0 \).
2. **Statement #2**: "The equation has three real solutions because there are three turning points on the graph."
- This statement is also false. The number of turning points does not directly correlate with the number of solutions. A polynomial can have more turning points than solutions.
3. **Statement #3**: "The equation has four real solutions because there are four x-intercepts."
- This statement is true. The equation being a polynomial of degree 4 can have up to 4 real solutions, and having four x-intercepts indicates that there are indeed four real solutions.
### Conclusion
Thus, the correct statement is:
**Statement #3 is true.**
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