5.3 Graphing Polynomial Functions

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What is a polynomial function?

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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is: f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

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What does the degree of a polynomial indicate?

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The degree of a polynomial is the highest power of the variable in the polynomial. It indicates the maximum number of roots (or zeros) the polynomial can have.

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What is a root of a polynomial?

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A root of a polynomial is a value of x for which the polynomial evaluates to zero. It is also known as a zero of the polynomial.

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What is the significance of the leading coefficient in a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree. It affects the end behavior of the polynomial function.

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What does it mean for a graph to 'bounce' at a root?

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A graph 'bounces' at a root when the root has an even multiplicity. This means the graph touches the x-axis at that root but does not cross it.

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What does it mean for a graph to 'cross' at a root?

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A graph 'crosses' at a root when the root has an odd multiplicity. This means the graph intersects the x-axis at that root.

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How do you determine the end behavior of a polynomial function?

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The end behavior of a polynomial function can be determined by the degree and the leading coefficient. If the degree is even and the leading coefficient is positive, both ends of the graph rise. If the degree is odd and the leading coefficient is positive, the left end falls and the right end rises.

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What is the maximum number of turning points in a polynomial function?

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The maximum number of turning points in a polynomial function is one less than the degree of the polynomial.

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What is the Intermediate Value Theorem?

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The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different signs at the endpoints, then there exists at least one c in (a, b) such that f(c) = 0.

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How do you find the zeros of a polynomial?

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To find the zeros of a polynomial, set the polynomial equal to zero and solve for x. This may involve factoring, using the quadratic formula, or applying numerical methods.