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Find the following for the function f(x)=(x+6)^2(x-3)^2

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a) Find the x- and y - intercept of the polynomial function f.

b) Find the power function that  the graph of f resembles for large values of |x|.

c) Determine the maximum number of turning points on the graph of f.

d) Determine the behavior of the graph of f near each x-intercept
asked Sep 8, 2014 in ALGEBRA 2 by anonymous

4 Answers

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(a).

The function is f(x) = (x + 6)2(x - 3)2.

To find the y-intercept, substitute x = 0 in the original function.

f(0) = (0 + 6)2(0 - 3)2

      = (36)(9)

      = 324.

To find the x-intercept, we solve f(x) = 0.

(x + 6)2(x - 3)2 = 0

[(x + 6)(x - 3)]2 = 0

(x + 6)(x - 3) = 0

x + 6 = 0 and x - 3 = 0

x = - 6 and x = 3.

The x-intercepts are - 6 and 3.

answered Sep 8, 2014 by casacop Expert
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(d).

The function is f(x) = (x + 6)2(x - 3)2.

The two x-intercepts are - 6 and 3.

Near - 6 :

f(x) = (x + 6)2(- 6 - 3)2

f(x) = 81(x + 6)2

Since 81 > 0, the parabola open open up at x = - 6.

Near 3 :

f(x) = (3 + 6)2(x - 3)2

f(x) = 81(x + 6)2

Since 81 > 0, the parabola open open up at x = 3.

answered Sep 9, 2014 by casacop Expert
0 votes

(c).

The function is f(x) = (x + 6)2(x - 3)2.

Expand the polynomial to write it in the form

f(x) = anxn + an - 1 xn - 1 + · · · · + a1x + a0.

f(x) = [x2 +12x + 36] * [x2 - 6x + 9]

f(x) = x2[x2 - 6x + 9] + 12x[x2 - 6x + 9] + 36[x2 - 6x + 9]

f(x) = x4 - 6x3 + 9x2 + 12x3 - 72x2 + 108x + 36x2 - 216x + 324

f(x) = x4 + 6x3 - 27x2 - 108x + 324.

Because the polynomial function f(x) is of degree 4, the graph of the function will have at most 4 - 1 = 2 turning points.

 

answered Sep 9, 2014 by casacop Expert
0 votes

(b).

The function is f(x) = (x + 6)2(x - 3)2.

Determine the end behavior of the graph of the function.

Expand the polynomial to write it in the form

f(x) = anxn + an - 1 xn - 1 + · · · · + a1x + a0.

f(x) = [x2 +12x + 36] * [x2 - 6x + 9]

f(x) = x2[x2 - 6x + 9] + 12x[x2 - 6x + 9] + 36[x2 - 6x + 9]

f(x) = x4 - 6x3 + 9x2 + 12x3 - 72x2 + 108x + 36x2 - 216x + 324

f(x) = x4 + 6x3 - 27x2 - 108x + 324.

The polynomial function f(x) is of degree 4. The graph of f(x) behaves like y = x4 for large values of | x |.

answered Sep 9, 2014 by casacop Expert
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