Write each polynomial function in standard form, classify it by degree, and determine the end behavior of its graph.
y = 2x^3 - 6 + 3x^2 - x^4 + 12
The given polynomial function is:
y = 2x^3 - 6 + 3x^2 - x^4 + 12
Let's rearrange the terms in descending order of degree:
y = -x^4 + 2x^3 + 3x^2 - 6 + 12
The standard form of the polynomial function is:
y = -x^4 + 2x^3 + 3x^2 + 6
The degree of this polynomial is 4, as the highest power of x is 4.
To determine the end behavior of the graph, we consider the leading term, which is -x^4. Since the coefficient of the leading term is negative, as x approaches positive or negative infinity, the value of y will also approach negative infinity.
Hence, the end behavior of the graph is:
As x approaches positive or negative infinity, y approaches negative infinity.