Question

Analyze the polynomial function ​f(x)equalsnegative 4 (x plus 3 )(x minus 4 )cubed using parts​ (a) through​ (e).a) Determine the end behavior of the graph of the function.​(b) Find the​ x- and​ y-intercepts of the graph of the function.

Answer 1

Answer:

a) f(x) tends to minus infinity

b) (0,768) is the y-intercept, (4,0) and (-3,0) are the x-intercepts.

Step-by-step explanation:

Our function is $f(x)=-4(x+3)(x-4)^3$ a polynomial of degree 4.

a) The monomial with highest degree determines the behavior of f(x) when x tends to infinity and when x tends to -infinity. This monomial is -4x⁴. Without expanding completely, (x-4)³ has x³ as a summand, which multiplies with -4x from the first factor, to give -4x⁴. When x goes to infinity (or minus infinitive), x²=(x²)² is positive (nonzero squares are always positive) thus -4x² is negative, and f(x) tends to minus infinity.

b). To find the y-intercept, we compute (0,f(0)). Since f(0)=-4(3)(-4)³=768, then (0,768) is the point of the y-intercept.

For the x-intercept, solve f(x)=0. f(x)=0 has the solutions x=-3 and x=4. In this case, the x-intercept is in both x=0 and x=4. Then (-3,0) and (4,0) are x-intercepts.