graph the following polynomial function. factor first if the expression is not in factored form. use the…

graph the following polynomial function. factor first if the expression is not in factored form. use the rational zeros theorem as necessary. f(x) = 2x(x - 1)(x + 2). use the graphing tool to graph the function. click to enlarge graph

graph the following polynomial function. factor first if the expression is not in factored form. use the rational zeros theorem as necessary. f(x) = 2x(x - 1)(x + 2). use the graphing tool to graph the function. click to enlarge graph

Answer

Explanation:

Step1: Identify x-intercepts

Set ( f(x) = 0 ), so ( 2x(x - 1)(x + 2)=0 ). The solutions are ( x = 0 ), ( x = 1 ), ( x=-2 ). These are the x - intercepts.

Step2: Determine end behavior

The degree of the polynomial is ( 3 ) (since when expanded, the highest power of ( x ) is ( 3 )) and the leading coefficient is ( 2 ) (positive). For a polynomial of odd degree with positive leading coefficient, as ( x\rightarrow+\infty ), ( f(x)\rightarrow+\infty ); as ( x\rightarrow-\infty ), ( f(x)\rightarrow-\infty ).

Step3: Analyze multiplicity of roots

The root ( x = 0 ) has multiplicity ( 1 ), ( x = 1 ) has multiplicity ( 1 ), ( x=-2 ) has multiplicity ( 1 ). Since all multiplicities are odd, the graph crosses the x - axis at each of these points.

Step4: Find y - intercept

Set ( x = 0 ) in ( f(x) ), ( f(0)=2\times0\times(0 - 1)\times(0 + 2)=0 ). So the y - intercept is ( (0,0) ) (which we already knew from the x - intercept).

Step5: Test some additional points (optional but helpful)

For ( x = 0.5 ), ( f(0.5)=2\times0.5\times(0.5 - 1)\times(0.5 + 2)=1\times(-0.5)\times2.5=-1.25 ) For ( x=-1 ), ( f(-1)=2\times(-1)\times(-1 - 1)\times(-1 + 2)=(-2)\times(-2)\times1 = 4 ) For ( x = 2 ), ( f(2)=2\times2\times(2 - 1)\times(2 + 2)=4\times1\times4 = 16 ) For ( x=-3 ), ( f(-3)=2\times(-3)\times(-3 - 1)\times(-3 + 2)=(-6)\times(-4)\times(-1)=-24 )

Using these x - intercepts, end - behavior, and additional points, we can sketch the graph. The function is a cubic polynomial with roots at ( x=-2,0,1 ), crosses the x - axis at these points, has the end - behavior of an odd - degree polynomial with positive leading coefficient, and passes through the calculated points.

Answer:

To graph ( f(x)=2x(x - 1)(x + 2) ):

  1. x - intercepts: ( (-2,0) ), ( (0,0) ), ( (1,0) ) (graph crosses the x - axis here as multiplicities are odd).
  2. End - behavior: As ( x\rightarrow+\infty ), ( f(x)\rightarrow+\infty ); as ( x\rightarrow-\infty ), ( f(x)\rightarrow-\infty ).
  3. y - intercept: ( (0,0) ).
  4. Additional points (e.g., ( (0.5,-1.25) ), ( (-1,4) ), ( (2,16) ), ( (-3,-24) )) help in shaping the graph. Use a graphing tool (like a graphing calculator or software) to plot these features and draw the curve.