1.2 Describing end behavior of polynomial and rational functions

• Steps to determine the end behavior of a polynomial function

  1️⃣ Identify the degree of the polynomial

  2️⃣ Identify the leading coefficient

  3️⃣ Use the degree and leading coefficient to describe the end behavior
• Match the degree and leading coefficient with the correct end behavior:

  Even degree, Negative leading coefficient ↔️ Both ends approach -∞

  Odd degree, Negative leading coefficient ↔️ Left: ∞, Right: -∞
• What is the degree and leading coefficient of $3x^{4} - 5x^{2} +$$2x - 7$?


  Degree: 4, Coefficient: 3
• What determines the end behavior of a rational function?

  Horizontal asymptotes
• The end behavior of a rational function is determined by the presence of horizontal asymptotes
• When the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.


  True
• If the degree of a polynomial is even and the leading coefficient is positive, both ends approach positive infinity
• The degree of a polynomial is the highest power of x
• Describe the end behavior of the polynomial $f(x) =$$2x^{3} - 5x +$$1$.


  $f(x) \to - \infty$ as $x \to - \infty$ and $f(x) \to \infty$ as $x \to \infty$
• For $f(x) =$ \frac{3x^{2} + 1}{x^{2} - 4}, the horizontal asymptote is 3
• The degree of the denominator in a rational function is the highest power of x
• Match the degree and leading coefficient with the correct end behavior:

  Even, Positive ↔️ $f(x) \to \infty$ as $x \to \pm \infty$

  Even, Negative ↔️ $f(x) \to - \infty$ as $x \to \pm \infty$

  Odd, Positive ↔️ $f(x) \to - \infty$ as $x \to - \infty$ and $f(x) \to \infty$ as $x \to \infty$

  Odd, Negative ↔️ $f(x) \to \infty$ as $x \to - \infty$ and $f(x) \to - \infty$ as $x \to \infty$
• For the polynomial f(x) = 2x^{3} - 5x + 1</latex>, what is its degree?

  3
• What is the horizontal asymptote when the degree of the numerator is less than the degree of the denominator?

  $y =$$0$
• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y =$$0$.


  True
• For $f(x) =$ \frac{3x^{2} + 1}{x^{2} - 4}, the horizontal asymptote is $y =$$3$.


  True
• What is the degree of the numerator in a rational function?

  Highest power of $x$
• For polynomial functions, the end behavior depends on the degree and the leading coefficient.
• What is the end behavior of $f(x) =$$3x^{4} - 2x +$$1$?


  $f(x) \to \infty$ as $x \to \pm \infty$
• What happens to the end behavior of a rational function if the degree of the numerator is greater than the denominator?

  $f(x)$ increases or decreases without bound
• The end behavior of a polynomial function is determined by the leading coefficient and the degree
• What is the end behavior of a polynomial with an odd degree and a positive leading coefficient?

  Left: -∞, Right: ∞
• The leading coefficient is the coefficient in front of the term with the highest power of x
• A polynomial with an odd degree and a negative leading coefficient approaches positive infinity on the left and negative infinity on the right.

  True
• What does the end behavior of a rational function describe?

  Behavior as $x \to \pm \infty$
• What is the horizontal asymptote when the degrees of the numerator and denominator are equal?

  $y =$$\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$
• What determines the end behavior of a polynomial function?

  Degree and leading coefficient
• Match the degree and leading coefficient with the corresponding end behavior:

  Even degree, positive leading coefficient ↔️ $f(x) \to \infty$ as $x \to \pm \infty$

  Even degree, negative leading coefficient ↔️ $f(x) \to - \infty$ as $x \to \pm \infty$

  Odd degree, positive leading coefficient ↔️ $f(x) \to - \infty$ as $x \to - \infty$ and $f(x) \to \infty$ as $x \to \infty$
• For the polynomial $3x^{4} - 5x^{2} +$$2x - 7$, the degree is $4$ and the leading coefficient is $3$.


  True
• Identifying the degrees of the numerator and denominator is crucial for determining the end behavior of rational functions.


  True
• What is the degree of the numerator in a rational function?

  The highest power of x
• What is the end behavior of an even degree polynomial with a positive leading coefficient?

  $f(x) \to \infty$ as $x \to \pm \infty$
• The leading coefficient is the coefficient of the term with the highest power of $x$.


  True
• Match the degree comparison with the correct end behavior of rational functions:

  Numerator < Denominator ↔️ $f(x) \to 0$ as $x \to \pm \infty$

  Numerator = Denominator ↔️ $f(x) \to y$ as $x \to \pm \infty$

  Numerator > Denominator ↔️ $f(x) \to \infty$ or $f(x) \to - \infty$ as $x \to \pm \infty$
• What guides the end behavior of rational functions?

  Horizontal asymptotes
• What happens to the end behavior of a rational function if the degree of the numerator is greater than the degree of the denominator?

  $f(x) \to \infty$ or $f(x) \to - \infty$
• Match the degree comparison with the correct horizontal asymptote and end behavior:

  Numerator < Denominator ↔️ $y =$$0$ and $f(x) \to 0$

  Numerator = Denominator ↔️ $y =$$\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$ and $f(x) \to y$

  Numerator > Denominator ↔️ None and $f(x) \to \infty$ or $f(x) \to - \infty$
• If the degree of the numerator is less than the denominator, what is the horizontal asymptote?

  $y =$$0$
• If the degree of the numerator is less than the degree of the denominator, the end behavior is $f(x) \to 0$ as $x \to \pm \infty$.


  True
• What does the end behavior of a polynomial function describe?

  Function's behavior at infinity