Module 3: Lesson 1

 

Self-Check 4

2.  
   

   3. f(x) = −(x + 3)3(x − 4)

       

      Step 1: Determine the end behaviour by looking at the degree of the polynomial and the sign of the leading coefficient.
      
      There are four factors containing x; therefore, the expanded form of the polynomial is of degree 4. The degree of the polynomial is even and the polynomial has a negative leading coefficient. This means the graph will start in quadrant 3 and end in quadrant 4.

       

      Step 2: Factor the polynomial if it is not given in the problem.
      
      This question is already factored.

       

      Step 3: Determine the x-intercepts by looking at the factors of the polynomial.
      
      The x-intercepts are x = −3 and 4.

       

      Step 4: Determine the y-intercept by substituting zero into the function.
      
      The y-intercept occurs at the point (0, 108).

       

      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 3, so the graph will “flatten out” and cross the x-axis at x = −3. The factor (x − 4) has a multiplicity of 1, so the graph will cross the x-axis at x = 4.

       

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

       
      
      

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