Math Problem Statement
Solution
The graph shown in the image is smooth and continuous, without any sharp corners or breaks. Based on this, let's evaluate the choices:
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First Option:
- "Graph is not a polynomial because it has a jump."
- This is incorrect, as the graph does not have any jumps or discontinuities. It is smooth throughout.
- "Graph is not a polynomial because it has a jump."
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Second Option:
- "Graph is not a polynomial because it has a sharp turn."
- This is also incorrect, since the graph does not show any sharp turns (corners). Polynomials are smooth and continuous, and this graph fits that description.
- "Graph is not a polynomial because it has a sharp turn."
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Third Option:
- "Graph is a polynomial because it is continuous, domain is all reals, and it does not have any sharp turns."
- This is the correct option. The graph is continuous, smooth (no sharp turns), and extends across all real numbers in the domain. This is typical of polynomial functions.
- "Graph is a polynomial because it is continuous, domain is all reals, and it does not have any sharp turns."
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Fourth Option:
- "Graph is not a Polynomial because the domain is not all reals."
- This is incorrect, as the graph appears to have a domain over all real numbers (since it extends infinitely to the left and right).
- "Graph is not a Polynomial because the domain is not all reals."
Thus, the correct answer is the third option: the graph represents a polynomial because it is continuous, smooth (no sharp turns), and has a domain that includes all real numbers.
Would you like further details on polynomial functions, or have any additional questions?
Here are some related questions:
- What are the characteristics of a polynomial graph?
- How can you distinguish a polynomial from a non-polynomial graph?
- What are sharp turns in a graph, and why don’t polynomials have them?
- Why must polynomials be continuous across all real numbers?
- Can a graph with a discontinuity ever represent a polynomial function?
Tip: Polynomials are always smooth and continuous, so checking for jumps or sharp turns is a quick way to rule out non-polynomials.
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Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Continuity
Graph Behavior
Formulas
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Theorems
Polynomials are continuous functions without sharp turns.
Suitable Grade Level
Grades 9-12