Question

1. which of the following statements correctly describes the end behavior of the function below?

y = x² + 5x + 4
○ as the x-values approach negative infinity, the y-values approach positive infinity. as the x-values approach positive infinity, the y-values approach negative infinity.
○ as the x-values approach negative infinity, the y-values approach negative infinity. as the x-values approach positive infinity, the y-values approach positive infinity.
○ as the x-values approach negative infinity, the y-values approach positive infinity. as the x-values approach positive infinity, the y-values approach positive infinity.
○ as the x-values approach negative infinity, the y-values approach negative infinity. as the x-values approach positive infinity, the y-values approach negative infinity.

Explanation:

Step1: Identify the function type

The function given is \( y = x^2 + 5x + 4 \), which is a quadratic function. The general form of a quadratic function is \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In this case, \( a = 1 \), \( b = 5 \), and \( c = 4 \).

Step2: Analyze the leading coefficient

For a quadratic function, the end - behavior is determined by the leading coefficient (\( a \)) and the degree of the function (which is 2 for a quadratic). The degree of the function is even (2 is even), and the leading coefficient \( a = 1>0 \).

Step3: Determine the end - behavior based on degree and leading coefficient

When the degree of a polynomial function is even and the leading coefficient is positive:

• As \( x

ightarrow-\infty \) (as \( x \) approaches negative infinity), we consider the term with the highest power, which is \( ax^2 \). When \( x \) is a large negative number, \( x^2 \) is a large positive number (because squaring a negative number gives a positive result). Since \( a = 1>0 \), \( ax^2=x^2 \) will be a large positive number. So, as \( x
ightarrow-\infty \), \( y = x^2+5x + 4\approx x^2
ightarrow+\infty \) (the \( 5x \) and \( 4 \) terms become negligible compared to \( x^2 \) when \( |x| \) is very large).

• As \( x

ightarrow+\infty \) (as \( x \) approaches positive infinity), \( x^2 \) is a large positive number (since squaring a positive number gives a positive result) and \( a = 1>0 \). So, \( ax^2=x^2
ightarrow+\infty \), and again the \( 5x \) and \( 4 \) terms are negligible compared to \( x^2 \) when \( x \) is very large. So, as \( x
ightarrow+\infty \), \( y=x^2 + 5x+4\approx x^2
ightarrow+\infty \).

So the end - behavior of the function \( y=x^2 + 5x + 4\) is that as \( x \) approaches both negative infinity and positive infinity, \( y \) approaches positive infinity.

Answer:

The second option (the one stating "As the \( x \)-values approach negative infinity, the \( y \)-values approach positive infinity. As the \( x \)-values approach positive infinity, the \( y \)-values approach positive infinity.")