Graph each polynomial function. ƒ(x)=2x³+x²-x

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1. Identify the polynomial function given: \(f(x) = 2x^3 + x^2 - x\).
2. Find the x-intercepts by setting \(f(x) = 0\) and solving the equation \$2x^3 + x^2 - x = 0$. Factor the polynomial to find the roots.
3. Determine the y-intercept by evaluating \(f(0)\), which is the constant term or the value of the function when \(x=0\).
4. Find the critical points by computing the first derivative \(f'(x)\) and setting it equal to zero: \(f'(x) = \frac{d}{dx}(2x^3 + x^2 - x)\). Solve \(f'(x) = 0\) to find potential maxima, minima, or points of inflection.
5. Use the critical points and intercepts to sketch the graph, considering the end behavior of the cubic function (as \(x \to \pm \infty\), \(f(x) \to \pm \infty\) depending on the leading term).

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables raised to whole-number exponents and coefficients combined using addition, subtraction, and multiplication. Understanding the degree and leading coefficient helps predict the general shape and end behavior of the graph.

Graphing Cubic Functions

Cubic functions are polynomial functions of degree three, typically having an S-shaped curve. Key features include intercepts, turning points, and end behavior, which depend on the sign and magnitude of the leading coefficient.

Finding Intercepts and Critical Points

To graph a polynomial, find the x-intercepts by solving ƒ(x)=0 and the y-intercept by evaluating ƒ(0). Critical points, found by setting the derivative equal to zero, indicate local maxima or minima, helping to sketch the curve accurately.