Question

which of the following describes the function ( x^4 - 3 )?
the degree of the function is even, so the ends of the graph continue in opposite directions. because the leading coefficient is positive, the left side of the graph continues down the coordinate plane and the right side continues upward
the degree of the function is even, so the ends of the graph continue in the same direction. because the leading coefficient is negative, the left side of the graph continues downward and the right side also continues downward
the degree of the function is even, so the ends of the graph continue in opposite directions. because the leading coefficient is negative, the left side of the graph continues up the coordinate plane and the right side continues downward
the degree of the function is even, so the ends of the graph continue in the same direction. because the leading coefficient is positive, the left side of the graph continues up the coordinate plane and the right side also continues upward

Explanation:

Step1: Analyze the function's degree

The function is \( f(x)=x^4 - 3 \). The degree of a polynomial is the highest power of \( x \), so here the degree is 4, which is even. For even - degree polynomials, the end - behaviors of the left - hand side and the right - hand side of the graph are the same.

Step2: Analyze the leading coefficient

The leading term of the polynomial \( f(x)=x^4 - 3 \) is \( x^4 \), and the leading coefficient (the coefficient of the term with the highest power of \( x \)) is 1, which is positive.
For a polynomial with an even degree and a positive leading coefficient, as \( x
ightarrow+\infty \), \( f(x)=x^4 - 3
ightarrow+\infty \) (the right - hand end of the graph goes up), and as \( x
ightarrow-\infty \), since \( (-x)^4=x^4 \), \( f(x)=(-x)^4 - 3=x^4 - 3
ightarrow+\infty \) (the left - hand end of the graph also goes up). So the ends of the graph continue in the same direction, and since the leading coefficient is positive, both the left and right sides of the graph continue upward.

Answer:

The degree of the function is even, so the ends of the graph continue in the same direction. Because the leading coefficient is positive, the left side of the graph continues up the coordinate plane and the right side also continues upward (the fourth option in the given choices).