How do you use end behavior, zeros, y intercepts to sketch the graph of #g(x)=(x+1)(x-2)^2(x-4)#?

1 Answer
Nov 29, 2017

graph{(x+1)(x-2)^2(x-4) [-10, 10, -5, 5]}

Explanation:

This is a quartic function. The equation has degree 4, so it is an even- degree polynomial.

The leading coefficient is positive, so as #x# approaches negative infinity, the graph rises to the left, and as #x # approaches positive infinity, the graph rises to the right.

In order to solve for the #y#-intercept, replace all of the #x# variables with 0's.

#y = (0+1)(0-2)^2(0-4)#

#y = (1)(4)(-4)#

#y = -16#

The constant term is -16, so the coordinate of the #y#-intercept is (0,-16).

The zeroes of the function are the #x#-intercepts of the graph. The #x#-intercepts of this equation are -1, 2, and 4.

The zero -1 has multiplicity 1, so the graph crosses the #x#-axis at the related #x#-intercept.

The zero 2 has multiplicity 2, so the graph just touches the #x#-axis at the related #x#-intercept.

The zero 4 has multiplicity 1, so the graph crosses the #x#-axis at the related #x#-intercept.