2. sketch $f(x)=(x + 1)^{2}(x - 1)(x - 3)$.
2. sketch $f(x)=(x + 1)^{2}(x - 1)(x - 3)$.
Answer
Explanation:
Step1: Find the x - intercepts
Set (y = f(x)=0). By the zero - product property, if ((x + 1)^{2}(x - 1)(x - 3)=0), then (x=-1) (with multiplicity (m = 2)), (x = 1) (with multiplicity (m=1)), and (x = 3) (with multiplicity (m = 1)).
Step2: Determine the end - behavior
The leading term of (y=(x + 1)^{2}(x - 1)(x - 3)) is obtained by multiplying the leading terms of each factor. ((x+1)^{2}=x^{2}+2x + 1), and ((x^{2}+2x + 1)(x-1)(x - 3)=(x^{2}+2x + 1)(x^{2}-4x + 3)=x^{4}-4x^{3}+3x^{2}+2x^{3}-8x^{2}+6x+x^{2}-4x + 3=x^{4}-2x^{3}-4x^{2}+2x + 3). As (x\rightarrow\pm\infty), (y = f(x)\approx x^{4}). So, as (x\rightarrow-\infty), (y\rightarrow+\infty) and as (x\rightarrow+\infty), (y\rightarrow+\infty).
Step3: Analyze the behavior at the x - intercepts
- For (x=-1) (multiplicity (m = 2), even): The graph touches the x - axis at (x=-1). The derivative (y^\prime) (using the product rule (y=(x + 1)^{2}(x - 1)(x - 3)), (y^\prime=2(x + 1)(x - 1)(x - 3)+(x + 1)^{2}(x - 3)+(x + 1)^{2}(x - 1)). At (x=-1), (y^\prime=0).
- For (x = 1) (multiplicity (m = 1), odd): The graph crosses the x - axis at (x = 1).
- For (x = 3) (multiplicity (m = 1), odd): The graph crosses the x - axis at (x = 3).
Step4: Find the y - intercept
Set (x = 0). Then (y=(0 + 1)^{2}(0 - 1)(0 - 3)=(1)\times(-1)\times(-3)=3). So the y - intercept is ((0,3)).
Answer:
Plot the x - intercepts ((-1,0)), ((1,0)), ((3,0)), the y - intercept ((0,3)). Use the end - behavior (as (x\rightarrow\pm\infty), (y\rightarrow+\infty)) and the behavior at the x - intercepts (touch at (x=-1), cross at (x = 1) and (x = 3)) to sketch the graph of (y=(x + 1)^{2}(x - 1)(x - 3)).