Question

pick the graph of the following function.

\\(f(x) = (x - 1)(x - 4)^2\\)

Explanation:

Step1: Find the x-intercepts

$$x - 1 = 0 \implies x = 1$$

$$(x - 4)^2 = 0 \implies x = 4$$

Step2: Determine the multiplicity behavior

At \(x = 1\), multiplicity is \(1\) (crosses x-axis). At \(x = 4\), multiplicity is \(2\) (touches x-axis).

Step3: Determine the y-intercept

$$f(0) = (0 - 1)(0 - 4)^2 = -16$$

Step4: Identify the correct graph

The bottom-left graph crosses at \(x = 1\) and touches at \(x = 4\).

Answer:

The fourth graph (bottom-left graph) is the correct graph of the function.