Describe the behavior of the graph at the x-intercepts for the function f(x)=(2x-7)^7(x+3)^4. Identify each x-intercept and justify your answer as well.
To determine the x-intercepts of the function f(x)=(2x-7)^7(x+3)^4, we set f(x) equal to zero and solve for x.
(2x-7)^7(x+3)^4 = 0
Setting each factor equal to zero:
2x-7 = 0 --> x = 7/2
x+3 = 0 --> x = -3
So, there are two x-intercepts: x = 7/2 and x = -3.
Now, let's analyze the behavior of the graph at each x-intercept:
1. x = 7/2:
At x = 7/2, the function (2x-7)^7 equals zero since it is raised to the power of 7. Therefore, the contribution from this factor to the overall function f(x) will be zero.
However, the factor (x+3)^4 will still have an effect on the function since it is raised to the power of 4, and it is not equal to zero when x = 7/2. Therefore, the behavior of the graph at x = 7/2 will depend on the behavior of (x+3)^4. Since (x+3) is positive when x > -3, raising it to the power of 4 will still yield a positive value. Thus, the graph will not cross the x-axis or change sign at x = 7/2.
2. x = -3:
At x = -3, both factors, (2x-7)^7 and (x+3)^4, are equal to zero. Therefore, the behavior of the graph will be that it crosses the x-axis at x = -3.
In summary, the behavior of the graph at the x-intercept x = 7/2 is that it does not cross the x-axis or change sign, while the behavior at the x-intercept x = -3 is that it crosses the x-axis.