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Dec 13, 2025

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4 pages

Exploring Polynomial Functions

Franco Antonacci

@francoantonacci_qloc

End behavior describes how polynomial functions behave as x approaches... Show more

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1- End behavior.notebook
For polynomials whose degree is even:
If the coefficient of the leading term is positive, then
"y"
as x→∞, then f(x

End Behavior of Even-Degree Polynomials

When dealing with polynomials with an even degree (like x², x⁴), the ends of the graph will either both point upward or both point downward. The leading coefficient determines the direction:

If the leading coefficient is positive, both ends point upward. As x approaches positive infinity or negative infinity, y approaches positive infinity. We write this using limits: lim(x→∞) f(x) = ∞ and lim $x→-∞$ f(x) = ∞.

If the leading coefficient is negative, both ends point downward. As x approaches either positive or negative infinity, y approaches negative infinity. We write: lim(x→∞) f(x) = -∞ and lim $x→-∞$ f(x) = -∞.

💡 Quick Tip: For even-degree polynomials, think of a parabola (x²) shape - both ends go in the same direction, either both up (positive leading coefficient) or both down (negative leading coefficient).

Several example problems show how to determine the degree and leading coefficient of polynomials like 2x³-10x²+6x+22 (degree 3, leading coefficient 2) and ½x $x+2$ ² $x-3$ ³ $x+4$ (degree 7, leading coefficient ½).

End Behavior of Odd-Degree Polynomials

Odd-degree polynomials (like x³, x⁵) have ends pointing in opposite directions, creating a crossing graph. The leading coefficient determines which end points upward:

If the leading coefficient is positive, the right end points upward and the left end points downward. This means as x→∞, f(x)→∞ and as x→-∞, f(x)→-∞.

If the leading coefficient is negative, the right end points downward and the left end points upward. As x→∞, f(x)→-∞ and as x→-∞, f(x)→∞.

The page shows example problems where students identify end behavior and y-intercepts. For instance, f(x) = 2x⁴+x³-x²+5x+3 has degree 4 with positive leading coefficient, so both ends point upward, and its y-intercept is (0,3).

🔑 Remember: For odd-degree polynomials, the ends always point in opposite directions - one up and one down. The leading coefficient tells you which is which!

The y-intercept is always found by substituting x=0 into the function.

Analyzing End Behavior in Complex Polynomials

When dealing with factored polynomials like m(x) = 4 $x+3$ ² $x-5$ ³, you need to multiply out the highest power terms to find the degree and leading coefficient. This example has degree 5 with leading coefficient 4.

For polynomials in factored form, determining the y-intercept requires careful substitution. For m(x) = 4 $x+3$ ² $x-5$ ³, when x=0, we get 4(0+3)²(0-5)³ = 4(9)(-125) = -4500, giving y-intercept (0,-4500).

The page includes multiple-choice questions asking students to match functions with their graphs based on end behavior. For example, f(x) = 3x⁵-x³+4x+22 is an odd-degree polynomial with positive leading coefficient, so its right end points up and left end points down.

👉 Application Insight: When answering multiple-choice questions about polynomial graphs, first identify the degree (odd or even) and the sign of the leading coefficient to eliminate incorrect options before examining other features!

Understanding end behavior helps you quickly sketch reasonable polynomial graphs and recognize polynomial functions from their graphical representations.

Practice and Application

This page appears to be a continuation of practice problems or possibly a blank page for students to work through examples.

When practicing end behavior analysis, follow these steps:

Identify the degree of the polynomial (odd or even)
Find the sign of the leading coefficient (positive or negative)
Determine the direction of both ends of the graph
Calculate the y-intercept by evaluating f(0)

You can quickly rule out incorrect graph options on tests by checking if the end behavior matches your analysis. This skill will help you on both homework and exams!

💪 You've Got This: End behavior may seem abstract at first, but with practice, you'll be able to look at any polynomial and quickly visualize how its graph behaves at the extremes.

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Algebra 2

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203

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Dec 13, 2025

•

4 pages

Exploring Polynomial Functions

Franco Antonacci

@francoantonacci_qloc

End behavior describes how polynomial functions behave as x approaches positive or negative infinity. Understanding end behavior helps you predict what happens at the "far ends" of polynomial graphs without having to plot every point.

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End Behavior of Even-Degree Polynomials

When dealing with polynomials with an even degree (like x², x⁴), the ends of the graph will either both point upward or both point downward. The leading coefficient determines the direction:

💡 Quick Tip: For even-degree polynomials, think of a parabola (x²) shape - both ends go in the same direction, either both up (positive leading coefficient) or both down (negative leading coefficient).

Sign up to see the contentIt's free!

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End Behavior of Odd-Degree Polynomials

Odd-degree polynomials (like x³, x⁵) have ends pointing in opposite directions, creating a crossing graph. The leading coefficient determines which end points upward:

If the leading coefficient is positive, the right end points upward and the left end points downward. This means as x→∞, f(x)→∞ and as x→-∞, f(x)→-∞.

If the leading coefficient is negative, the right end points downward and the left end points upward. As x→∞, f(x)→-∞ and as x→-∞, f(x)→∞.

🔑 Remember: For odd-degree polynomials, the ends always point in opposite directions - one up and one down. The leading coefficient tells you which is which!

The y-intercept is always found by substituting x=0 into the function.

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Analyzing End Behavior in Complex Polynomials

👉 Application Insight: When answering multiple-choice questions about polynomial graphs, first identify the degree (odd or even) and the sign of the leading coefficient to eliminate incorrect options before examining other features!

Understanding end behavior helps you quickly sketch reasonable polynomial graphs and recognize polynomial functions from their graphical representations.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

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Practice and Application

This page appears to be a continuation of practice problems or possibly a blank page for students to work through examples.

When practicing end behavior analysis, follow these steps:

Identify the degree of the polynomial (odd or even)
Find the sign of the leading coefficient (positive or negative)
Determine the direction of both ends of the graph
Calculate the y-intercept by evaluating f(0)

You can quickly rule out incorrect graph options on tests by checking if the end behavior matches your analysis. This skill will help you on both homework and exams!

💪 You've Got This: End behavior may seem abstract at first, but with practice, you'll be able to look at any polynomial and quickly visualize how its graph behaves at the extremes.