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 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 local low point.

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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 local high point.

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How do you find relative extrema of a polynomial function?

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

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What is the end behavior of polynomial functions?

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The end behavior of a polynomial function 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 end behavior of the polynomial function.

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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 consists of terms with non-negative integer exponents, while a rational function is a ratio of two polynomials.

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What is the significance of the x-intercepts of a polynomial function?

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The x-intercepts of a polynomial function are the points where the graph crosses the x-axis. They represent the solutions to the equation f(x) = 0.