1. sketch ( f(x)=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2} ).

1. sketch ( f(x)=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2} ).

1. sketch ( f(x)=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2} ).

Answer

Explanation:

Step1: Find the x - intercepts

Set (f(x)=0). Then ((x + 3)^{3}(x + 4)^{2}(x - 2)^{2}=0). Using the zero - product property (a\times b\times c = 0) implies (a = 0) or (b=0) or (c = 0). (x+3=0) gives (x=-3); (x + 4=0) gives (x=-4); (x-2=0) gives (x = 2). The x - intercepts are ((-3,0)), ((-4,0)), and ((2,0)).

Step2: Determine the multiplicity and the behavior at the x - intercepts

  • For (x=-3) with multiplicity (n = 3) (odd). As (x\to - 3^{-}), (y=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2}\approx(-)^{3}(+)^{2}(+)^{2}=-). As (x\to - 3^{+}), (y=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2}\approx(+)^{3}(+)^{2}(+)^{2}=+). The graph crosses the x - axis at (x=-3).
  • For (x=-4) with multiplicity (n = 2) (even). As (x\to - 4^{-}), (y=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2}\approx(+)^{3}(-)^{2}(+)^{2}=+). As (x\to - 4^{+}), (y=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2}\approx(+)^{3}(+)^{2}(+)^{2}=+). The graph touches the x - axis at (x=-4).
  • For (x = 2) with multiplicity (n=2) (even). As (x\to2^{-}), (y=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2}\approx(+)^{3}(+)^{2}(-)^{2}=+). As (x\to2^{+}), (y=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2}\approx(+)^{3}(+)^{2}(+)^{2}=+). The graph touches the x - axis at (x = 2).

Step3: Determine the end - behavior

The leading term is obtained by multiplying the leading terms of each factor. ((x + 3)^{3}\approx x^{3}), ((x + 4)^{2}\approx x^{2}), ((x - 2)^{2}\approx x^{2}). Then (f(x)\approx x^{3}\times x^{2}\times x^{2}=x^{7}). As (x\to-\infty), (y=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2}\approx x^{7}\to-\infty). As (x\to+\infty), (y=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2}\approx x^{7}\to+\infty).

To sketch the graph:

  • Plot the x - intercepts ((-4,0)), ((-3,0)), and ((2,0)).
  • Use the behavior at the intercepts (cross at (x=-3), touch at (x=-4) and (x = 2)).
  • Use the end - behavior ((y\to-\infty) as (x\to-\infty) and (y\to+\infty) as (x\to+\infty)).

Answer:

Follow the steps above to sketch the graph of (y=(x + 3)^{3}(x + 4)^{2}(x - 2)^{2}) with x - intercepts at ((-4,0)), ((-3,0)), ((2,0)), crossing the x - axis at (x=-3) and touching the x - axis at (x=-4) and (x = 2), and having end - behavior (y\to-\infty) as (x\to-\infty) and (y\to+\infty) as (x\to+\infty).