Questions and Answers
Which graph shows the same end behavior as the graph of f(x) = 2x^6 - 2x^2 - 5?
Which of the following graphs could be the graph of the function f(x) = x^4 + x^3 - x^2 - x?
What is the end behavior of the graph of the polynomial function f(x) = 2x^3 - 26x - 24?
At which root does the graph of f(x) = (x + 4)^6(x + 7)^5 cross the x-axis?
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Which of the following graphs could be the graph of the function f(x)= -0.08x(x² - 11x + 18)?
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Which statement describes the graph of f(x) = -x^4 + 3x^3 + 10x^2?
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What is the end behavior of the graph of the polynomial function f(x) = 3x^6 + 30x^5 + 75x^4?
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Let a and b be real numbers where a ≠ b ≠ 0. Which of the following functions could represent the graph below?
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At which root does the graph of f(x) = (x - 5)^3(x + 2)^2 touch the x-axis?
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Which statement describes the graph of f(x) = -4x^3 - 28x^2 - 32x + 64?
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Which of the following graphs could be the graph of the function f(x)= 0.03x²(x² - 25)?
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A polynomial function has a root of -5 with multiplicity 3, a root of 1 with multiplicity 2, and a root of 3 with multiplicity 7. If the function has a negative leading coefficient and is of even degree, which statement about the graph is true?
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Which graph has the same end behavior as the graph of f(x) = -3x^3 - x^2 + 1?
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Which statement describes the graph of f(x) = 4x^7 + 40x^6 + 100x^5?
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A polynomial function has a root of -4 with multiplicity 4, a root of -1 with multiplicity 3, and a root of 5 with multiplicity 6. If the function has a positive leading coefficient and is of odd degree, which could be the graph of the function?
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Which of the following graphs could be the graph of the function f(x)= 0.003x^2(x^2 - 25)?
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Study Notes
Graphing Polynomial Functions
- End behavior of polynomial functions can be determined by examining the leading term.
- For the function f(x) = 2x^6 - 2x^2 - 5, graph A shows the same end behavior; graph D does not.
Graphs of Polynomial Functions
- The function f(x) = x^4 + x^3 - x^2 - x corresponds to graph A.
End Behavior of Specific Polynomial
- The polynomial f(x) = 2x^3 - 26x - 24 has end behavior where, as x approaches negative infinity, y approaches negative infinity, and as x approaches positive infinity, y approaches positive infinity.
Roots and the X-Axis
- The graph of f(x) = (x + 4)^6(x + 7)^5 crosses the x-axis at the root -7.
Graph Representation
- The function f(x) = -0.08x(x² - 11x + 18) aligns with graph C.
Intercepts of Polynomial Graphs
- For the function f(x) = -x^4 + 3x^3 + 10x^2, the graph touches the x-axis at x = 0 and crosses at x = 5 and x = -2.
End Behavior Analysis
- The polynomial f(x) = 3x^6 + 30x^5 + 75x^4 exhibits end behavior where both ends go to positive infinity as x approaches both negative and positive infinity.
Function Representation
- A function f(x) that could represent the described graph is f(x) = (x - a)^2(x - b)^4.
X-Axis Intersections
- In the function f(x) = (x - 5)^3(x + 2)^2, the x-axis is touched at the root -2.
Crossing and Touching the X-Axis
- For f(x) = -4x^3 - 28x^2 - 32x + 64, the graph touches the x-axis at x = -4 and crosses at x = 1.
Graphing Functions
- The function f(x) = 0.03x²(x² - 25) corresponds to graph A.
Multiplicity of Roots
- A polynomial with roots of -5 (multiplicity 3), 1 (multiplicity 2), and 3 (multiplicity 7), with a negative leading coefficient and an even degree, behaves negatively on the interval (3, ∞).
End Behavior of Specific Polynomial
- The function f(x) = -3x^3 - x^2 + 1 has its end behavior matching that of graph D.
X-Axis Behavior
- The function f(x) = 4x^7 + 40x^6 + 100x^5 crosses the x-axis at x = 0 and touches it at x = -5.
Graph Representation for Odd Degrees
- For a polynomial with roots of -4 (multiplicity 4), -1 (multiplicity 3), and 5 (multiplicity 6), a positive leading coefficient and odd degree indicates it corresponds to graph B.
Graph Identification
- The function f(x) = 0.003x^2(x^2 - 25) aligns with graph A.
Studying That Suits You
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Test your understanding of polynomial functions and their graphs with these flashcards. Each card challenges you to identify end behavior and graph characteristics for various polynomial functions. Perfect for students preparing for math exams or anyone looking to refresh their skills.
