Math Problem Statement
Solution
We are tasked with identifying the correct graph of the polynomial function based on the provided conditions:
Given Information:
- is positive on intervals and .
- is negative on intervals and .
- is increasing on intervals and .
- is decreasing on interval .
Key Analysis:
-
Sign of :
- is positive in the intervals and , which means the graph should be above the x-axis in these intervals.
- is negative in the intervals and , so the graph should be below the x-axis in these ranges.
-
Increasing/Decreasing Behavior:
- is increasing on and . This suggests that the slope of the graph is upward in these intervals.
- is decreasing on , meaning the slope is downward in this region.
Matching with the Graphs:
Based on these observations, we can eliminate graphs that do not follow the sign and behavior changes correctly:
- Graph a: Shows a positive value between and , negative value in and , and the correct increasing/decreasing pattern.
- Graph b, c, and d do not match all the required increasing/decreasing patterns and signs.
Correct Answer:
Graph a matches all the given conditions.
Do you want more details about the analysis or have any further questions?
Here are some related questions to deepen understanding:
- How do we determine when a function is increasing or decreasing on a graph?
- What role do critical points play in determining the behavior of a polynomial function?
- How can the sign of help in sketching polynomial functions?
- How do turning points affect the shape of polynomial graphs?
- What other features of polynomials (like concavity) could help in graph sketching?
Tip: When analyzing graphs, always track both the signs and slopes of the function to understand its behavior over different intervals.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Intervals
Increasing and Decreasing Functions
Positive and Negative Values
Formulas
-
Theorems
Intermediate Value Theorem (implicitly used to identify where the function changes signs)
Suitable Grade Level
Grades 10-12
Related Recommendation
Analyze Polynomial Graph: Intervals of Increase and Decrease
Analyzing the Polynomial Function from a Graph
Graphing Polynomial Functions Based on Given Conditions
Identify Increasing Intervals for a Function Using Graph Analysis
Analyzing Relationships in Functions: Increasing, Decreasing, and Positive/Negative Intervals