Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=-x(x+1)(x-1)

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Identify the given polynomial function: \(f(x) = -x(x+1)(x-1)\). Notice that it is already factored into three linear factors.

Recognize the zeros of the function by setting each factor equal to zero: \(x = 0\), \(x + 1 = 0 \Rightarrow x = -1\), and \(x - 1 = 0 \Rightarrow x = 1\). These are the x-intercepts of the graph.

Determine the end behavior of the polynomial. Since the leading term comes from multiplying \(-x\), \(x\), \(x+1\), and \(x-1\), the degree is 3 (odd degree) and the leading coefficient is negative, so as \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).

Plot the zeros on the x-axis at \(x = -1\), \(0\), and \(1\). Then, choose a test point in each interval determined by these zeros to find the sign of \(f(x)\) in those intervals, which helps to sketch the curve between the intercepts.

Use the information about zeros, end behavior, and test points to sketch the graph of \(f(x)\). Remember the graph crosses the x-axis at each zero because each factor is to the first power.

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주요 개념

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Factoring Polynomial Functions

Factoring involves expressing a polynomial as a product of its factors, which simplifies analysis and graphing. For example, the given function is already factored as -x(x+1)(x-1), showing its roots clearly. Factoring helps identify zeros and simplifies evaluating the function.

Zeros of a Polynomial Function

Zeros are the values of x where the polynomial equals zero, found by setting each factor equal to zero. For ƒ(x) = -x(x+1)(x-1), the zeros are x = 0, -1, and 1. These points are where the graph crosses or touches the x-axis, crucial for sketching the graph.

End Behavior of Polynomial Functions

End behavior describes how the graph behaves as x approaches positive or negative infinity, determined by the leading term's degree and sign. Here, the leading term is -x³, so as x → ±∞, ƒ(x) → ∓∞, meaning the graph falls to the right and rises to the left. Understanding this guides the overall shape of the graph.