Given the function: f(x)=(x-1)(x-2)^2 please answer the following for questions

Question

Given the function: f(x)=(x-1)(x-2)^2 please answer the following for questions

1. List intervals determined by the x-intercepts.

 

2. Select at least one x value in each interval and show how it indicates if the graph is positive or negative within that interval.

 

3. Use the results you found to sketch the graph of the function. Use dashed lines to show the intervals. 

Answer 1

1)

 

To find the x-intercepts, we set f(x) equal to zero.

 

0 = (x - 1)(x - 2)²

 

 

From this equation,

 

x = 1          and         x = 2

 

 

2)

 

The two x values from the previous question are on the x-axis.  Therefore, the other x values are either below or above the x-axis.  Below x-axis means f(x) is negative.  Above x-axis means f(x) is positive.

 

Let's evaluate f(x) when  x=0 , x=1.5 , and x=3.   When we do so, f(x) will either be a positive or negative value. 

 

 

f(0) = (0 - 1)(0 - 2)²

      = (-1)(-2)²

      = (-1)(4)

      = -4

 

f(1.5) = (1.5 - 1)(1.5 - 2)²

         = (0.5)(-0.5)²

         = 0.125

 

f(3) = (3 - 1)(3 - 2)²

       = (2)(1)²

       = 2

 

 

This means the f(x) is negative in the interval  (-∞, 1).

This means that f(x) is positive in the interval (1, 2)∪(2, ∞).

 

 

3)

 

When f(x) is expanded, we get a degree of 3.  So f(x) will act like a  x³ function.  You probably know f(x) = x³   looks like.  f(x) is also positive, so it starts off increasing and ends increasing.

 

The graph crosses the x-axis at (0, 1).  The graph touches the x-axis at (0, 2), but never crosses it at this specific point. 

 

The y-intercept is (0, -4).

 

 

Using this information, can you sketch the graph?