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

Question

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

Answer 1

(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.

Answer 2

(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.

Answer 3

(c).

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

Expand the polynomial to write it in the form

f(x) = a_nxⁿ + a_{n - 1} x^{n - 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.

 

Answer 4

(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) = a_nxⁿ + a_{n - 1} x^{n - 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 |.

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