Please help with steps

Question

Consider the function f(x)=x³+x²-4x-4 to determine each of the following:

a). The function's end behavior

b). The x-intercept(s).Make sure to clearly state whether the graph crosses the x-axis ORtouches the x-axis and turns around at each intercept.

c). The y-intercept

d). Does the graph have y-axis symmetry, origin symmetry, or neither? Make sure to ex[lain.

Answer 1

a) the highest degree term dominates at the end behavior. x^3 > 0, so as x approaches infinity, x^3 approaches infinity. when x approaches negative infinity, x^3 approaches negative infinity. For very large x, x^3 gets increasingly large. For very negative x, x^3 gets increasingly negative.

b) x^3+x^2-4x-4=0 has a zero or x intercept at x=-1 (-1)^3 +(-1)^2 -4(-1) -4 =0.

-1+1 +4-4 =0 (x+1) is a factor. divide x+1 into x^3+x^2 -4x -4 to get a quadratic factor: x^2-4. It factors further: (x-2)(x+2). That means x=2 and x=-2 are two more roots or zeros. x intercepts are 2, -2 and -1. the curves crosses the y axis at three points: (2,0), (-2,0) and (-1,0). The curve fully intersects the x axis. It does not just touch and turn around at any of these points. The turn around points are where the slope = 0. take the derivative to find those points. y'= 3x^2 +2x = 0. x(3x+2) =0. the curves turns around at x=0 and x=-2/3. Those are local extrema. (0,-4) is a local or relative minimum. (-2/3,0) is a local or relative maximum.

c) y intercept is y=-4 or the point (0,-4) where the curve intersects the y axis.

just set x = to zero and solve for y.

d) check to see if f(x) = -f(-x) or if f(x) = f(-x). Compare f(x)=

x^3 +x^2 -4x -4 with (-x)^3 + (-x)^2 -4(-x) -4 = -x^3 +x^2 +4x -4

f(x) does not = f(-x). And f(x) does not = -f(x). It almost does except for the x^2 term. There is neither symmetry about the origin or the y axis