draw sketch of graph of function described

Question

the degree of the function is 6.lead coefficient is negative. Zeros: -3 multiplicity is 2, -1 multiplicity is 1 and 2 multiplicity of 3

Answer 1

The first term of the function is -x⁶.  

 

The multiplicity of root -3 is 2, you will have two roots of -3.  The factor is (x + 3)². 

The multiplicity of root -1 is 1.  The factor is (x + 1).

The multiplicity of root 2 is 3.  The factor is (x - 2)³.

 

Notice that when we expand these factors, the product of the first terms is x⁶.

 

When the value of the multiplicity is even, the graph is tangential to the x-axis at the respective root.  When the value is odd, then the graph crosses the x-axis. 

 

Since the degree of the leading term is 6, you will have 5 local extremas. The factor (x + 3)² represents a function similar to x².  The shape of the graph is a concave up parabola.  The factor (x + 1) represents a linear function.  The factor (x - 2)³ represents the graph similar to x³  which is not symmetrical to the y-axis as you are probably aware of. 

 

So at point (-3,0), you will draw a parabola that is concave up and tangential at that point.  At point (-1,0), you will draw a straight line that crosses the point.  Connect the endpoint of the parabola to point (-1,0). Then from point (-1,0), draw a line that represents the function similar to x³ and crosses the point (2,0). 

 

Also, the graph will have a y-intercept.  Let's plug in x=0 into the factors.

 

f(x) = (x + 3)²(x + 1)(x - 2)³

f(0) = (3)²(1)(-2)³

f(0) = 9(-8)

f(0) = -72

 

The y-intercept is (0, -72).

 

Since the leading term is negative, the graph starts off decreasing and finishes off increasing from left to right.