Find the following function
f(x)=(x+3)^2(x-1)^2
(a)Find the x and y intercept of the polynomial function f.
(b) Determine whether the graph of the f crosses or touches the x-axis at each x intercept.
(c) Put all the information to obtain the graph of f.
To find the x and y intercepts of the polynomial function f(x) = (x+3)^2(x-1)^2, we need to solve for x when the function equals zero.
(a) X-intercepts: To find the x-intercepts, let f(x) = 0 and solve for x.
Setting f(x) = 0, we have:
(x+3)^2(x-1)^2 = 0
Since the product of factors is zero if and only if at least one of the factors is zero, we can set each factor equal to zero and solve for x:
(x+3)^2 = 0
(x-1)^2 = 0
Solving for x in each equation, we have:
x+3 = 0
x-1 = 0
From these equations, we find x = -3 and x = 1. These are the x-intercepts of the function f(x).
Y-intercept: To find the y-intercept, we need to evaluate f(x) when x = 0.
Substituting x = 0 into the function, we get:
f(0) = (0+3)^2(0-1)^2 = 3^2(-1)^2 = 9*1 = 9
Therefore, the y-intercept is 9, which is the value of f(0).
(b) To determine whether the graph of f(x) crosses or touches the x-axis at each x-intercept, we need to analyze the multiplicity of each x-intercept.
The multiplicity of an x-intercept is determined by the power to which the factor is raised in the polynomial function.
In our case, the factor (x+3) is squared, and the factor (x-1) is also squared.
When the multiplicity of an x-intercept is even (in this case, squared), the graph touches the x-axis but does not cross it.
Therefore, the graph of f(x) touches the x-axis at x = -3 and x = 1.
(c) To obtain the graph of f(x), we can use the information we have gathered so far.
We know that f(x) has x-intercepts at x = -3 and x = 1. It touches the x-axis at both of these points.
Moreover, we found the y-intercept to be 9, so the point (0, 9) lies on the graph of f(x).
We also know the graph touches the x-axis but does not cross it at the x-intercepts.
To draw the complete graph of f(x), you can plot these points (-3, 0), (1, 0), and (0, 9) and then sketch a smooth curve passing through the points while keeping in mind that the graph touches but does not cross the x-axis at the x-intercepts.