Question

For each function: (1) determine the real zeros and state the multiplicity of any repeated zeros, (2) describe the end behavior, and (3) sketch the graph

1) f(x) = -x^4 + 8x^3 - 20x^2 + 16x

Answer 1

i have no clue i need this one too

Answer 2

The function f(x) = -x^4 + 8x^3 - 20x^2 + 16x has real zeros at x = 0, x = 2 (multiplicity 2), and x = 4 (multiplicity 1). The end behavior shows the graph declining at both ends due to the negative even-degree leading term. The graph can be sketched by plotting the zeros, considering multiplicity, and the end behavior.

Step-by-step explanation:

For the function f(x) = -x^4 + 8x^3 - 20x^2 + 16x, we first need to find the real zeros by factoring if possible or using numerical methods such as the Rational Root Theorem or synthetic division. In this case, we can factor by grouping:

• f(x) = x(-x^3 + 8x^2 - 20x + 16)
• f(x) = x(x - 2)^2(x - 4)

The real zeros are x = 0, x = 2 (with multiplicity 2) and x = 4 (with multiplicity 1).

The end behavior of the function can be determined from the leading term, which is -x^4. Since the coefficient of the leading term is negative and the degree is even, both ends of the graph will point downwards as x approaches positive and negative infinity.

To sketch the graph, we can plot the zeros and consider the multiplicity of each zero: a zero with multiplicity 2 will touch the x-axis and turn back, whereas a zero with multiplicity 1 will cross the x-axis. We should also keep in mind the end behavior, with the graph heading downwards as x approaches infinity in both directions.