Flashcard on Sketching Polynomial Graphs

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What is a polynomial function?

Back

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 $$a_n, a_{n-1}, ..., a_0$$ are constants and $$n$$ is a non-negative integer.

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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 polynomial's behavior and the maximum number of roots it can have.

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What is the leading coefficient of a polynomial?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It influences the end behavior of the graph.

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How do you determine the x-intercepts of a polynomial function?

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To find the x-intercepts of a polynomial function, set the function equal to zero and solve for x. The solutions are the x-intercepts.

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What is the significance of the roots of a polynomial?

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The roots (or zeros) of a polynomial are the values of x that make the polynomial equal to zero. They correspond to the x-intercepts of the graph.

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What is the difference between even and odd degree polynomials in terms of graph behavior?

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Even degree polynomials have graphs that rise on both ends, while odd degree polynomials have graphs that rise on one end and fall on the other.

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What does it mean for a root to have multiplicity?

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Multiplicity refers to the number of times a particular root appears in a polynomial. A root with even multiplicity touches the x-axis, while a root with odd multiplicity crosses it.

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How can you identify 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 leading coefficient. For example, if the leading coefficient is positive and the degree is even, the graph rises on both ends.

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What is the vertex of a polynomial graph?

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The vertex is the highest or lowest point on the graph of a polynomial function, depending on whether it opens upwards or downwards.

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What is the role of the y-intercept in a polynomial function?

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The y-intercept is the point where the graph crosses the y-axis, found by evaluating the polynomial at x=0.