Question

How do you find the end behavior and state the possible number of x intercepts and the value of the y intercept given `y=x^5-2x^4-3x^3+5x^2+4x-1`?

Answer 1

Graphical method reveals x-intercepts ( y = 0 ) : `--1.175`. nearly, `-1`, exactly, and 0.205, nearly. y-intercept ( x = 0 ) : `-1`.. As `x to +-oo, y to +-oo`.

Explanation:

As the sum of the coefficients in `y(-x) = 0, -1` is an x-intercept.

The first graph indicates end behavior of

`y to oo in`Q_1`as`x to oo#, and likewise,

`y to - oo` in `Q_3`, as `x to -oo`.

The second gives first approximations to the three x-intercept, of

which -1 is exact.

The next improves the positive intercept to 0.205.

The other negative intercept is improved to -1.15, using root-

bracketing method.

The last is for f', giving four turning points as zeros of f'.

graph{x^5-2x^4-3x^3+5x^2+4x-1 [-12.49, 12.49, -6.21, 6.28]}

graph{x^5-2x^4-3x^3+5x^2+4x-1 [-2.5, 2.5, -1.25, 1.25]}

graph{x^5-2x^4-3x^3+5x^2+4x-1 [.204 .206, -1.25, 1.25]}

graph{5x^4-8x^3-9x^2+10x+4 [-2.122, 2.121, -1.06, 1.062]}