Math Problem Statement
Find the open interval(s) where the following function is increasing, decreasing, or constant. Express your answer in interval notation.
Solution
The graph shown appears to be a parabola that opens upwards, representing a quadratic function. We can determine where the function is increasing or decreasing based on the behavior of the graph.
Analyzing the Graph:
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Decreasing Interval: The function is decreasing as the graph slopes downwards from the left side towards the vertex. This happens from to the vertex at .
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Increasing Interval: The function starts increasing after the vertex at , as the graph slopes upwards. This happens from to .
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Constant Interval: There is no interval where the function is constant because the graph is either increasing or decreasing everywhere.
Intervals in Notation:
- The function is decreasing on the interval .
- The function is increasing on the interval .
- There is no constant interval.
Would you like any further explanation or details on this? Here are some related questions you might find useful:
- How do you find the vertex of a quadratic function algebraically?
- What is the importance of the derivative in determining increasing and decreasing intervals?
- Can a quadratic function have more than one increasing and decreasing interval?
- How does the leading coefficient of a quadratic equation affect the direction of the parabola?
- What are the implications of symmetry in the graph of a quadratic function?
Tip: The vertex of a parabola is a critical point where the function changes from increasing to decreasing or vice versa.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Intervals
Graph Interpretation
Formulas
Quadratic function: f(x) = ax^2 + bx + c
Theorems
Vertex theorem
Properties of quadratic functions
Suitable Grade Level
Grades 9-11
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