Question

Graph the function. \(f(x)=x^2(x+1)^3(x-2)\)

Short answer

The function \(f(x)=x^2(x+1)^3(x-2)\) graph has x-intercepts at x=0, x=-1, and x=2. The function tends to positive infinity as x approaches either positive or negative infinity. There are relative maximas and minimas in between the x-intercepts. The graph of the function is consistent with these characteristics.

Step-by-step solution

Identify the x-intercepts

First set each factor of the function to zero and solve for \(x\). Therefore \(x=0, -1, 2\) are the x-intercepts.

Identify the end behavior

As \(x\) tends to positive or negative infinity, the function \(f(x)\) tends to positive infinity because it is a polynomial function with even degree, and its leading coefficient is positive.

Identify the relative maximas and minimas

The intervals, determined by the x-intercepts, are \((-∞, -1),\), \((-1, 0), (0, 2),\) and \((2, ∞)\). By testing a number from each interval in \(f'(x)\), determine where the function is increasing and where it is decreasing. Switches from increasing to decreasing correspond to relative maximas, while switches from decreasing to increasing correspond to relative minimas.

Graph the function

Plot the x-intercepts, asymptotes, local maximas, and minimas on the coordinate plane. Connect these points smoothly making sure that the graph above is consistent with the identified end behavior. Remember the graph must pass through all x-intercepts.

Key concepts

End Behavior

The end behavior of a polynomial function describes what happens to the function values as the input, or the variable "x," approaches positive or negative infinity. For the function \(f(x) = x^2(x+1)^3(x-2)\), the highest degree of any term in the expanded polynomial is 6 (as found by summing the exponents: 2 from \(x^2\), 3 from \((x+1)^3\), and 1 from \((x-2)\)). Since this degree is even, we can predict that both ends of the graph of the function will head towards positive infinity as \(x\) moves to extreme positive or negative values.

• As \(x \to \infty\), \(f(x) \to \infty\)
• As \(x \to -\infty\), \(f(x) \to \infty\)

This is because even-degree polynomial functions tend to have the same behavior at both ends, depending on the sign of the leading coefficient, which is positive in this case.

X-intercepts

X-intercepts of a polynomial function are the points where the graph crosses or touches the x-axis. To find the x-intercepts of the function \(f(x) = x^2(x+1)^3(x-2)\), we set the function equal to zero and solve for \(x\).

• \(x = 0\)
• \(x = -1\)
• \(x = 2\)

These values are found by solving \(x^2 = 0\), \((x+1)^3 = 0\), and \((x-2) = 0\). The x-intercept at \(x = 0\) is actually a point where the graph touches and rebounds off the x-axis, due to the factor \(x^2\). The x-intercept at \(x = -1\) behaves similarly due to the term \((x+1)^3\), indicating it will also rebound since it has an odd-powered exponent, meaning this point represents a change in direction on the x-axis, forming a curve. At \(x = 2\), the factor \((x-2)\) is linear, and thus the graph simply crosses the x-axis here.

Maximum and Minimum Points

In polynomial functions, maximum and minimum points are found between x-intercepts and within sections where the function changes from increasing to decreasing or vice versa. These indicate the highest and lowest points locally on the graph.

For the function \(f(x) = x^2(x+1)^3(x-2)\), consider the intervals established by the x-intercepts:

• \((-\infty, -1)\)
• \((-1, 0)\)
• \((0, 2)\)
• \((2, \infty)\)

Evaluate these regions using test points in the derivative \(f'(x)\). If the derivative is positive, the function increases in that interval; if negative, it decreases.

The transition from increasing to decreasing indicates a local maximum, while decreasing to increasing suggests a local minimum. By testing each interval, you can determine location and nature of such points, giving the graph its characteristic waves. However, without explicit computations, these points are not given exactly but guide the drawing of the curve.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting critical features mentioned above: x-intercepts, maxima, minima, and understanding end behavior.

• Start by plotting x-intercepts: \(x = 0, -1, 2\).
• Note that between these points and intervals, the function may rise or fall according to its end behavior.
• Identify local maxima and minima from analysis of \(f'(x)\), and use them to define hills and valleys.
• Finally, sketch the curve smoothly through these key points.

Make sure to double-check that your graph meets the end behavior expectations: in this case, both ends rising to infinity. By carefully considering these elements, you create a graph that accurately represents the polynomial function's behavior across its entire domain.