Sketch the graph of y = (x - 3)^4 (x + 1)^3 (x - 1) by finding its intercepts an

Question

Sketch the graph of y = (x - 3)^4 (x + 1)^3 (x - 1) by finding its intercepts and its limits as x rightarrow infinity and x rightarrow -infinity. The y-intercept is f(0) = (-3)^4 ()^2 (-1) = and the x-intercepts are found by setting y = 0: x = 3, 1, Notice that since (x-3)^4 is positive, the function doesn't change sign at 3: When x is large positive, all three factors are positive, so lim_x rightarrow infinity (x - 3)^4 (x + 1)^2 (x - 1) = infinity. When x is large negative, the first factor is large positive and the second and third factors are both large negative so lim_x rightarrow - infinity (x - 3)^4 (x + 1)^2 (x - 1) = infinity. Combining this information, we give a rough sketch of the graph in the figure.

Explanation / Answer

TO find the y intercept we have to plug x=0

f(0) = (0-3)4(0+1)2(0-1) = (-3)4(1)(-1) = -81

To find the x intercepts we have to plug f(x) =0 and solve for x

(x-3)4(x+1)2(x-1)=0

x-3=0 , x+1 =0 , x-1=0

x=3 , x=-1 , x=1

And from the graph we can say that the graph crosses the x axis at x=-1 and x=1

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