Question

graph the polynomial function f(x)=(x - 1)(x + 4)^2 using parts (a) through (e). (e) use the information to draw a complete graph of the function. use the graphing tool to graph the function.

Explanation:

Step1: Find the x - intercepts

Set \(f(x)=0\). So \((x - 1)(x + 4)^{2}=0\). Then \(x-1 = 0\) gives \(x = 1\) and \((x + 4)^{2}=0\) gives \(x=-4\). The x - intercepts are \(x = 1\) and \(x=-4\).

Step2: Determine the multiplicity of roots

The root \(x = 1\) has multiplicity 1 (since the factor \((x - 1)\) has exponent 1), and the root \(x=-4\) has multiplicity 2 (since the factor \((x + 4)\) has exponent 2). A root with odd multiplicity 1 crosses the x - axis and a root with even multiplicity 2 touches the x - axis.

Step3: Find the y - intercept

Set \(x = 0\). Then \(f(0)=(0 - 1)(0 + 4)^{2}=(-1)\times16=-16\). The y - intercept is \(y=-16\).

Step4: Determine the end - behavior

The leading term of the polynomial \(f(x)=(x - 1)(x + 4)^{2}=(x - 1)(x^{2}+8x + 16)=x^{3}+8x^{2}+16x-x^{2}-8x - 16=x^{3}+7x^{2}+8x - 16\) is \(x^{3}\). Since the leading coefficient is 1 (positive) and the degree is 3 (odd), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\).

To draw the graph, plot the x - intercepts at \(x = 1\) and \(x=-4\), the y - intercept at \(y=-16\), consider the end - behavior and the behavior at the x - intercepts (cross at \(x = 1\) and touch at \(x=-4\)).

Answer:

The steps above provide the necessary information to graph the function \(y=(x - 1)(x + 4)^{2}\) using a graphing tool.