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 n is a non-negative integer.

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial f(x) = 2x^3 + 3x^2 + 4, the degree is 3.

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What is a relative maximum?

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A relative maximum is a point on the graph of a function where the function value is higher than the values of the function at nearby points. It represents a peak in the graph.

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What is a relative minimum?

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A relative minimum is a point on the graph of a function where the function value is lower than the values of the function at nearby points. It represents a trough in the graph.

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How do you find relative maxima and minima?

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To find relative maxima and minima, take the derivative of the function, set it to zero to find critical points, and use the second derivative test or the first derivative test to determine if these points are maxima or minima.

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What is the end behavior of polynomial functions?

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The end behavior of polynomial functions describes how the function behaves as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

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What is the leading coefficient?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It influences the direction of the graph as x approaches infinity.

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What is a turning point?

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A turning point is a point on the graph of a function where the function changes direction from increasing to decreasing or vice versa. It corresponds to relative maxima and minima.

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What is the difference between a polynomial and a rational function?

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A polynomial function is a function that can be expressed as a polynomial, while a rational function is a ratio of two polynomial functions. Rational functions can have asymptotes, while polynomial functions do not.

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CCSS.HSF-IF.C.7D

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What is the significance of the x-intercepts?

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The x-intercepts of a polynomial function are the points where the graph crosses the x-axis. They represent the values of x for which the function equals zero.