Math Problem Statement
Solution
To analyze the graph of the polynomial function in the image and determine the intervals where the function is positive and negative, we need to observe where the graph is above or below the x-axis.
Positive Intervals:
A function is positive when the graph is above the x-axis (i.e., ).
From the graph:
- The function is positive between and .
- The function becomes positive again from to infinity.
Thus, the positive intervals are:
Negative Intervals:
A function is negative when the graph is below the x-axis (i.e., ).
From the graph:
- The function is negative between and .
- The function is negative again between and .
Thus, the negative intervals are:
Final Answer:
- Positive Intervals: ,
- Negative Intervals: ,
Would you like more details or have any further questions? Here are five related questions that might help deepen your understanding:
- How can you identify zeros of the function from a graph?
- What happens to the sign of a polynomial function when it crosses the x-axis?
- How can you determine if the function is increasing or decreasing on certain intervals?
- How would you express the intervals if the graph touched the x-axis but didn’t cross it?
- What kind of polynomial function might produce this kind of graph?
Tip: Always observe where the function intersects the x-axis to determine the exact points where the sign changes from positive to negative or vice versa.
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Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Graph Interpretation
Interval Notation
Formulas
-
Theorems
Polynomial Root Theorem
Suitable Grade Level
Grades 10-12
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