Analyzing Polynomial Graphs

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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 are the zeros of a polynomial?

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The zeros of a polynomial are the values of x for which the polynomial function equals zero. They are also known as the roots of the polynomial.

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How do you find the relative maximum of a polynomial graph?

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To find the relative maximum, identify the highest point in a given interval of the graph. This can be done by analyzing the first derivative of the function and finding critical points.

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What does it mean for a graph to be increasing?

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A graph is increasing on an interval if, as x increases, the value of f(x) also increases. This is indicated by a positive slope.

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What does it mean for a graph to be decreasing?

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A graph is decreasing on an interval if, as x increases, the value of f(x) decreases. This is indicated by a negative slope.

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What is the significance of the y-intercept in a polynomial graph?

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The y-intercept is the point where the graph crosses the y-axis, which occurs when x = 0. It represents the value of the polynomial function at that point.

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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. It determines the polynomial's end behavior and the maximum number of zeros.

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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 direction of the graph as x approaches positive or negative infinity.

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What is a turning point in a polynomial graph?

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

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How can 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 leading coefficient. If the degree is even, both ends of the graph will point in the same direction; if odd, they will point in opposite directions.