Question

write describe what the end behavior of a graph is and how it is determined. the end behavior of a graph describes the select choice values as the select choice values approach negative and positive infinity. it can be determined by select choice

Explanation:

To answer this, we first recall what end - behavior of a graph means:

Step 1: Define End - Behavior

The end - behavior of a graph describes the \( y \) - values (or the function values, usually denoted as \( f(x) \)) as the \( x \) - values approach negative and positive infinity (\( x
ightarrow-\infty \) and \( x
ightarrow+\infty \)).

Step 2: Determine How to Find It

For a polynomial function \( y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0 \), the end - behavior is determined by the leading term \( a_nx^n \) (the term with the highest degree \( n \)) and the leading coefficient \( a_n \). The degree \( n \) (whether it is even or odd) and the sign of the leading coefficient \( a_n \) (positive or negative) tell us what happens to the \( y \) - values as \( x \) approaches \( \pm\infty \). For example, if we have a polynomial \( y = 2x^3+3x^2 - 5x+1 \), the leading term is \( 2x^3 \). Since the degree \( 3 \) is odd and the leading coefficient \( 2>0 \), as \( x
ightarrow+\infty \), \( y
ightarrow+\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow-\infty \).

For non - polynomial functions like rational functions, exponential functions, or logarithmic functions, we analyze the behavior of the function as \( x \) approaches \( \pm\infty \) by looking at the dominant terms in the numerator and denominator (for rational functions), the base of the exponential function (for exponential functions), etc. But in the context of a typical high - school or introductory college math course, when talking about end - behavior, the most common case is for polynomial functions, and we use the leading term (degree and leading coefficient) to determine it.

So, to fill in the blanks:

1. The end behavior of a graph describes the \( \boldsymbol{y} \) (or function) values as the \( \boldsymbol{x} \) values approach negative and positive infinity.
2. It can be determined by the \( \boldsymbol{leading\ term} \) (or the degree and leading coefficient for polynomials, or the dominant terms/characteristics of the function for non - polynomials) of the function (especially for polynomial functions, the leading term \( a_nx^n \) where \( n \) is the degree and \( a_n \) is the leading coefficient).

Answer:

To answer this, we first recall what end - behavior of a graph means:

Step 1: Define End - Behavior

The end - behavior of a graph describes the \( y \) - values (or the function values, usually denoted as \( f(x) \)) as the \( x \) - values approach negative and positive infinity (\( x
ightarrow-\infty \) and \( x
ightarrow+\infty \)).

Step 2: Determine How to Find It

For a polynomial function \( y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0 \), the end - behavior is determined by the leading term \( a_nx^n \) (the term with the highest degree \( n \)) and the leading coefficient \( a_n \). The degree \( n \) (whether it is even or odd) and the sign of the leading coefficient \( a_n \) (positive or negative) tell us what happens to the \( y \) - values as \( x \) approaches \( \pm\infty \). For example, if we have a polynomial \( y = 2x^3+3x^2 - 5x+1 \), the leading term is \( 2x^3 \). Since the degree \( 3 \) is odd and the leading coefficient \( 2>0 \), as \( x
ightarrow+\infty \), \( y
ightarrow+\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow-\infty \).

For non - polynomial functions like rational functions, exponential functions, or logarithmic functions, we analyze the behavior of the function as \( x \) approaches \( \pm\infty \) by looking at the dominant terms in the numerator and denominator (for rational functions), the base of the exponential function (for exponential functions), etc. But in the context of a typical high - school or introductory college math course, when talking about end - behavior, the most common case is for polynomial functions, and we use the leading term (degree and leading coefficient) to determine it.

So, to fill in the blanks:

1. The end behavior of a graph describes the \( \boldsymbol{y} \) (or function) values as the \( \boldsymbol{x} \) values approach negative and positive infinity.
2. It can be determined by the \( \boldsymbol{leading\ term} \) (or the degree and leading coefficient for polynomials, or the dominant terms/characteristics of the function for non - polynomials) of the function (especially for polynomial functions, the leading term \( a_nx^n \) where \( n \) is the degree and \( a_n \) is the leading coefficient).