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 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 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 is 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 is 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 to zero to find critical points, and use the second derivative test or the first derivative test to determine if they are minima or maxima.

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What is the significance of the first derivative in graphing polynomial functions?

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The first derivative of a polynomial function indicates the slope of the tangent line to the graph. It helps identify increasing and decreasing intervals and locate relative extrema.

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What is the significance of the second derivative in graphing polynomial functions?

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The second derivative indicates the concavity of the graph. If the second derivative is positive, the graph is concave up; if negative, it is concave down. It also helps confirm whether critical points are minima or maxima.

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What does it mean for a polynomial function to be continuous?

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A polynomial function is continuous if there are no breaks, jumps, or holes in its graph. This means you can draw the graph without lifting your pencil.

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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 and its degree.

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What is a turning point in a polynomial function?

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A turning point is a point on the graph where the function changes direction from increasing to decreasing or vice versa. The number of turning points is at most one less than the degree of the polynomial.