Question

graph the polynomial function f(x)=x^3 - 4x. answer parts (a) through (e). (a) determine the end behavior of the graph of the function. the graph of f behaves like y = x^3 for large values of |x|. (b) find the x - and y - intercepts of the graph of the function. the x - intercept(s) is/are . (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)

Explanation:

Step1: Find x - intercepts

Set \(y = f(x)=0\), so \(x^{3}-4x = 0\). Factor out \(x\): \(x(x^{2}-4)=0\). Then factor \(x^{2}-4\) as \((x + 2)(x - 2)\) using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\) where \(a=x\) and \(b = 2\). We get \(x(x + 2)(x - 2)=0\). By the zero - product property, if \(abc = 0\), then \(a=0\) or \(b = 0\) or \(c = 0\). So \(x=0\) or \(x=-2\) or \(x = 2\).

Step2: Find y - intercepts

Set \(x = 0\) in \(y=f(x)\). Then \(y=0^{3}-4\times0=0\).

Answer:

The x - intercept(s) is/are \(0,-2,2\)