analyzethe graph of the function
Find the x- and y-intercepts.
(b) Determine the end behavior: find the power function that the graph of f resembles for large values of |x|.
(c) Find the maximum number of turning points.
(d) Graph the function
Please show all of your work.
f(x)=-(x-1)(x+1)^3
y-intercepts (when x=0): (0,1)
x-intercepts: (-1,0) and (1,0)
end behavior: y decreases forever; approaches y = -x^4
Turning points: f is of degree 4, so a max of 3 turning points
In fact, it only has one at x = 1/2
To analyze the graph of the function f(x)=-(x-1)(x+1)^3 and find the x- and y-intercepts, determine the end behavior, find the maximum number of turning points, and graph the function, you can follow these steps:
Step 1: To find the x-intercepts, set f(x) equal to zero and solve for x:
-(x - 1)(x + 1)^3 = 0
The x-intercepts occur when either factor is equal to zero. So, we have two possible x-intercepts:
x - 1 = 0 ==> x = 1
or
x + 1 = 0 ==> x = -1
Therefore, the x-intercepts are x = 1 and x = -1.
Step 2: To find the y-intercept, set x equal to zero and evaluate f(x):
f(0) = -(0 - 1)(0 + 1)^3
f(0) = -(-1)(1)^3
f(0) = -(-1)(1)
f(0) = 1
The y-intercept is (0, 1).
Step 3: To determine the end behavior, analyze the power of the leading term in the polynomial. In this case, the function f(x) has a power of 4, which means it is a quartic function. For large values of |x|, the graph will resemble the behavior of a power function with the same power, which is a quartic function.
Step 4: To find the maximum number of turning points, consider the degree of the polynomial, which is the highest power in the polynomial. In this case, the degree is 4. The maximum number of turning points for a quartic function is n - 1, where n is the degree. So, the maximum number of turning points in this case is 4 - 1 = 3.
Step 5: To graph the function, start by plotting the x- and y-intercepts found earlier: (1, 0), (-1, 0), and (0, 1).
Next, consider the behavior of the function between and beyond these points. Since f(x) is a negative function (-), the graph will be below the x-axis for x < -1. Between -1 and 1, the graph will be above the x-axis. For x > 1, the graph will be below the x-axis again.
Considering these points and the end behavior, you can sketch the graph accordingly. Remember that the graph will have 3 turning points and resemble a quartic function for large values of |x|.
It's important to note that without specific instructions regarding the scale and range of the graph, the above steps provide a general guide for understanding and graphing the function. Adjustments might be necessary based on the context or specific requirements.