Module 3: Lesson 1

 

Self-Check 4

2.  
   

   2. f(x) = (x + 3)2(x − 2)

      
      

      Step 1: Determine the end behaviour by looking at the degree of the polynomial and the sign of the leading coefficient.
      
      There are three factors containing x; therefore, the expanded form of the polynomial is of degree 3. The degree of the polynomial is odd and the polynomial has a positive leading coefficient. This means the graph will start in quadrant 3 and end in quadrant 1.

       

      Step 2: Factor the polynomial if it is not given in the problem.
      
      The factored form is given: f(x) = (x + 3)2(x − 2) or f(x) = (x + 3)(x + 3)(x − 2).

       

      Step 3: Determine the x-intercepts by looking at the factors of the polynomial.
      
      The x-intercepts are x = −3 and 2.

       

      Step 4: Determine the y-intercept by substituting zero into the function.
      
      The y-intercept occurs at the point (0, −18).

       

      Step 5: Determine the nature of the x-intercepts (whether there is a sign change) by looking at the multiplicity of the polynomial’s factors.
      
      The factor (x + 3) has a multiplicity of 2, so the graph will not cross the x-axis at x = −3. The factor (x − 2) has a multiplicity of 1, so the graph will cross the x-axis at x = 2.

       

      Step 6: Draw a smooth curve through the x- and y-intercepts.
      
      

       
      
      

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