Question

Determining end behavior and intercepts to graph a polynomial function.Also for part (a) it asks it it falls to the left rises right or falls right rises left or falls both or rises both.

Answer 1

In this problem, we must analyze the behaviour of the function:

$f(x)=(x+2)^2\cdot(x-1)^2.$

(a) Plotting the function, we get the following graph:

We see that the end behaviour of function is: rises on both sides.

(b) By looking at the expression of the polynomial f(x), we see that it has:

• a zero at x = -2 with multiplicity 2,

,

• a zero at x = 1 with multiplicity 2.

The graph a polynomial has the following behaviour according to its zeros:

Using this data, we conclude that the function:

• do not cross the x-axis,

,

• touches but do not cross the x-axis at x = -2, 1.

(c) The y-intercept is the value of y = f(0), the value of f(x) when x = 0:

$f(0)=(0+2)^2\cdot(0-1)^2=4\cdot1=4.$

(d) The function has:

• zeros of order two at x = -2 and x = 1, so it touches but does not cross the a-axis there,

,

• y-intercept at y = 4.

The graph of the function and the x-intercept and y-intercept points is:

Answer

(a) rises on both sides

(b)

• do not cross the x-axis

,

• touches but do not cross the x-axis at x = ,-2, 1

(c) y-intercept = 4