Podcast
Questions and Answers
What is the degree of the polynomial $4x^2(x + 2)(x - 5)^3$?
What is the degree of the polynomial $4x^2(x + 2)(x - 5)^3$?
- 5
- 6 (correct)
- 7
- 4
For a polynomial with a negative leading coefficient and an odd degree, what is its end behavior?
For a polynomial with a negative leading coefficient and an odd degree, what is its end behavior?
- Extends down to the left, and up to the right
- Extends down to the left, and down to the right
- Extends up to the left, and down to the right (correct)
- Extends up to the left, and up to the right
How does the multiplicity of a zero affect the graph at that x-intercept?
How does the multiplicity of a zero affect the graph at that x-intercept?
- The multiplicity does not affect the graph at the x-axis.
- An odd multiplicity causes the graph to be tangent to the x-axis.
- An even multiplicity causes the graph to be tangent to the x-axis. (correct)
- An even multiplicity causes the graph to cross the x-axis.
Consider the polynomial with factors of $(x-3)^2$ and $(x+1)^3$. What are the zeros and their multiplicities?
Consider the polynomial with factors of $(x-3)^2$ and $(x+1)^3$. What are the zeros and their multiplicities?
How can the y-intercept of a polynomial be determined?
How can the y-intercept of a polynomial be determined?
What is the maximum number of turning points a polynomial function of degree 5 can have?
What is the maximum number of turning points a polynomial function of degree 5 can have?
If a polynomial has a root at x = 3 with a multiplicity of 2, what behavior will the graph exhibit at x=3?
If a polynomial has a root at x = 3 with a multiplicity of 2, what behavior will the graph exhibit at x=3?
What is the domain of the cubic parent function $f(x) = x^3$?
What is the domain of the cubic parent function $f(x) = x^3$?
If a polynomial has a root with multiplicity 1, what happens at that x-intercept?
If a polynomial has a root with multiplicity 1, what happens at that x-intercept?
A polynomial has a relative minimum at x=a. What is true about the function values around that point?
A polynomial has a relative minimum at x=a. What is true about the function values around that point?
How does the graph of a polynomial function behave at a root with a multiplicity of 3?
How does the graph of a polynomial function behave at a root with a multiplicity of 3?
What does the term 'turning point' refer to on the graph of a polynomial function?
What does the term 'turning point' refer to on the graph of a polynomial function?
What does the term 'multiplicity' refer to in the context of polynomial equations?
What does the term 'multiplicity' refer to in the context of polynomial equations?
How many x-intercepts can a polynomial function of degree 'n' have?
How many x-intercepts can a polynomial function of degree 'n' have?
A relative minimum on the graph is best described as:
A relative minimum on the graph is best described as:
Which statement is true about a polynomial function's 'zero'?
Which statement is true about a polynomial function's 'zero'?
What is the relationship between the absolute maximum or minimum of a polynomial function and its range?
What is the relationship between the absolute maximum or minimum of a polynomial function and its range?
What role does a graphing calculator play when analyzing a graph (as mentioned in the content)?
What role does a graphing calculator play when analyzing a graph (as mentioned in the content)?
When graphing a polynomial, what does 'end behavior' describe?
When graphing a polynomial, what does 'end behavior' describe?
How does a 'relative maximum' differ from an 'absolute maximum'?
How does a 'relative maximum' differ from an 'absolute maximum'?
If a root has a multiplicity of 2, what does this indicate about the graph at that x-intercept?
If a root has a multiplicity of 2, what does this indicate about the graph at that x-intercept?
Flashcards
Multiplicity
Multiplicity
The number of times a root appears in a polynomial equation. For example, a root with a multiplicity of 2 means the graph touches the x-axis at that point but doesn't cross it.
Relative Minimum
Relative Minimum
The lowest point on a small section of the graph. The y-value at this point is a relative minimum.
Relative Maximum
Relative Maximum
The highest point on a small section of the graph. The y-value at this point is a relative maximum.
Absolute Minimum/Maximum
Absolute Minimum/Maximum
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Zero (of a function)
Zero (of a function)
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Degree of a Polynomial
Degree of a Polynomial
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Leading Coefficient
Leading Coefficient
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End Behavior
End Behavior
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Odd multiplicity
Odd multiplicity
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Even multiplicity
Even multiplicity
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Degree
Degree
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Turning point
Turning point
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Zeros
Zeros
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Zeros of a polynomial
Zeros of a polynomial
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Multiplicity of a zero
Multiplicity of a zero
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Study Notes
Warm-Up
- Determine the leading coefficient, degree, end behavior, maximum number of U-turns, and y-intercept for the following functions:
- f(x) = 8x⁵ – 7x³ + 3x – 7
- f(x) = –5(x – 2)³(3x + 5)²
- f(x) = –2x⁴(x² + 2)³(2x – 1)²
Characteristics of Polynomial Functions
- Use tables, graphs, and verbal descriptions to interpret key characteristics of a function that models the relationship between two quantities.
- Sketch a graph showing intercepts, intervals of increasing/decreasing/positive/negative, relative maximums/minimums, symmetries, and end behavior.
Vocabulary
- Multiplicity: The number of times a root occurs in a polynomial equation.
- Relative Minimum: The y-value at the lowest point on the graph.
- Relative Maximum: The y-value at the highest point on the graph.
- Absolute Minimum/Maximum: The highest or lowest point on the entire graph; defines the range.
- Zero: The x-value when f(x) = 0. Graphically, the x-intercepts.
Turning Points
- A turning point is where a graph changes from increasing to decreasing or vice versa.
- A turning point corresponds to a relative minimum or maximum.
- A polynomial function of degree n has at most n x-intercepts and at most (n – 1) turning points.
Relative Minimum/Maximum Values
- Relative minimum: f(a) is a relative minimum if, for all x in an open interval containing a, f(x) ≥ f(a).
- Relative maximum: f(a) is a relative maximum if, for all x in an open interval containing a, f(x) ≤ f(a).
Multiplicity
- If a root's multiplicity is odd, the graph crosses the x-axis at that value.
- If the root's multiplicity is 1, the graph crosses in a linear fashion.
- If the root's multiplicity is greater than 1, the graph crosses in a cubic fashion.
- If a root's multiplicity is even, the graph touches the x-axis at that value.
The Cubic Parent Function
- The function f(x) = x³ has the following characteristics:
- End Behavior: as x → -∞, f(x) → -∞; as x → ∞, f(x) → ∞
- Domain: (-∞, ∞)
- Range: (-∞, ∞)
- Increase: (-∞, ∞)
- Decrease: never
- Minimum: none
- Maximum: none
- Zeros: x = 0
- Turning Points: A graph changes direction at a turning point and it corresponds to a relative minimum or maximum.
Examples of Graphing Polynomials
- Detailed examples (1-5) are provided of how to sketch polynomials based on factors, degrees, and leading coefficients. Showing end behaviors, zeros, and y-intercepts.
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