Questions: Analyze the polynomial function f(x)=x^3+0.2x^2-1.4183x-0.41111. Complete parts (a) through (h). (a) Determine the end behavior of the graph of the function. The graph of f behaves like y=x for large values of x. (b) Graph the function using a graphing utility. Use the window [-3,3,1] by [-2,2,1]. Choose the correct graph below. (c) Use a graphing utility to approximate the x- and y-intercepts of the graph. The y-intercept is (Type an integer or decimal rounded to five decimal places as needed.)

Solution Steps

• The given polynomial function is \( f(x) = -0.2x^4 + 1.1483x^3 - 0.4111 \).
• The leading term is \( -0.2x^4 \).
• Since the leading coefficient is negative and the degree is even, the end behavior is:
  • As \( x \to \infty \), \( f(x) \to -\infty \).
  • As \( x \to -\infty \), \( f(x) \to -\infty \).

• Use a graphing calculator or software to plot the function \( f(x) = -0.2x^4 + 1.1483x^3 - 0.4111 \).
• Set the window to \([-3, 3]\) for \(x\) and \([-2, 2]\) for \(y\).
• Compare the graph with the given options to choose the correct one.

• Use the graphing utility to find the intercepts.
• The y-intercept is found by evaluating \( f(0) \): \[ f(0) = -0.2(0)^4 + 1.1483(0)^3 - 0.4111 = -0.4111 \]
• The x-intercepts are the points where the graph crosses the x-axis. Use the graphing utility to approximate these values.

Final Answer