Polynomial graph characteristics

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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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ 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 indicates the highest power of the variable in the polynomial. It determines the polynomial's end behavior and the 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 direction of the graph as x approaches positive or negative infinity.

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How can you determine if a polynomial function is even or odd?

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A polynomial is even if \(f(-x) = f(x)\) for all x, and odd if \(f(-x) = -f(x)\). Even degree polynomials have symmetric graphs about the y-axis, while odd degree polynomials have symmetric graphs about the origin.

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What is a relative maximum in a polynomial graph?

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A relative maximum is a point on the graph 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 in a polynomial graph?

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A relative minimum is a point on the graph 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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What is an absolute maximum?

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An absolute maximum is the highest point over the entire domain of the function. It is the largest value that the function can attain.

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What is an absolute minimum?

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An absolute minimum is the lowest point over the entire domain of the function. It is the smallest value that the function can attain.

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What does it mean for a polynomial function to be increasing on an interval?

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A polynomial function is increasing on an interval if, for any two points \(x_1\) and \(x_2\) in that interval where \(x_1 < x_2\), the function value at \(x_1\) is less than the function value at \(x_2\) (i.e., \(f(x_1) < f(x_2)\)).

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What does it mean for a polynomial function to be decreasing on an interval?

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A polynomial function is decreasing on an interval if, for any two points \(x_1\) and \(x_2\) in that interval where \(x_1 < x_2\), the function value at \(x_1\) is greater than the function value at \(x_2\) (i.e., \(f(x_1) > f(x_2)\)).