zuai-logo
zuai-logo
  1. AP Pre Calculus
FlashcardFlashcard
Study GuideStudy GuideQuestion BankQuestion Bank

What are the differences between polynomial and rational functions?

Polynomials: No breaks or asymptotes. | Rational Functions: Can have vertical/horizontal asymptotes and holes.

Flip to see [answer/question]
Flip to see [answer/question]
Revise later
SpaceTo flip
If confident

All Flashcards

What are the differences between polynomial and rational functions?

Polynomials: No breaks or asymptotes. | Rational Functions: Can have vertical/horizontal asymptotes and holes.

What are the differences between vertical asymptotes and holes in rational functions?

Vertical Asymptotes: Function approaches infinity. | Holes: Function is undefined, but limit exists.

What are the differences between even and odd degree polynomial functions regarding end behavior?

Even Degree: Ends go in the same direction. | Odd Degree: Ends go in opposite directions.

Compare and contrast linear and quadratic functions in terms of their rates of change.

Linear: Constant rate of change (slope). | Quadratic: Rate of change varies; not constant.

What is the difference between a zero of multiplicity 1 and a zero of multiplicity 2 in a polynomial function?

Multiplicity 1: The graph crosses the x-axis at the zero. | Multiplicity 2: The graph touches the x-axis and turns around at the zero.

Compare horizontal asymptotes when the degree of the numerator is less than, equal to, or greater than the degree of the denominator.

Numerator < Denominator: y=0. | Numerator = Denominator: y = ratio of leading coefficients. | Numerator > Denominator: No horizontal asymptote.

Compare end behavior of polynomials with positive and negative leading coefficients.

Positive Leading Coefficient: Graph rises to the right. | Negative Leading Coefficient: Graph falls to the right.

What is the difference between real and complex zeros of a polynomial function?

Real Zeros: Intersect the x-axis. | Complex Zeros: Do not intersect the x-axis.

Compare finding zeros of a polynomial by factoring versus using the quadratic formula.

Factoring: Useful for simple polynomials. | Quadratic Formula: Used when factoring is not possible (specifically for quadratics).

Compare the graphs of rational functions with and without holes.

Without Holes: Continuous except at asymptotes. | With Holes: Discontinuity at the location of the hole.

What is a polynomial function?

A sum of terms, each consisting of a coefficient and a variable raised to a non-negative integer power.

What is the degree of a polynomial?

The highest power of the variable in the polynomial.

What is a rational function?

A ratio of two polynomial functions, expressed as f(x)=p(x)/q(x)f(x) = p(x)/q(x)f(x)=p(x)/q(x).

What is a vertical asymptote?

A vertical line x=ax = ax=a where the function approaches infinity or negative infinity as xxx approaches aaa.

What is a horizontal asymptote?

A horizontal line y=by = by=b that the function approaches as xxx approaches infinity or negative infinity.

What is a hole in a rational function?

A point where the function is undefined because a factor cancels out in both the numerator and denominator.

Define rate of change.

How quickly a function's output changes with respect to its input.

What are complex zeros?

Zeros of a polynomial function that are complex numbers.

Define leading coefficient.

The coefficient of the term with the highest power in a polynomial.

What is end behavior?

The behavior of a function as xxx approaches positive or negative infinity.

Explain the concept of end behavior for polynomial functions.

End behavior describes what happens to the function's values as xxx approaches positive or negative infinity. It's determined by the leading term (degree and leading coefficient).

Explain how the degree of a polynomial affects its graph.

The degree determines the maximum number of turning points and the end behavior. Even degree: ends go in the same direction. Odd degree: ends go in opposite directions.

Explain how the leading coefficient of a polynomial affects its graph.

The sign of the leading coefficient determines whether the graph rises or falls as xxx approaches positive or negative infinity. Positive: rises to the right. Negative: falls to the right.

Explain the relationship between zeros and factors of a polynomial.

If x=ax = ax=a is a zero of a polynomial, then (x−a)(x - a)(x−a) is a factor of the polynomial.

Explain the concept of vertical asymptotes in rational functions.

Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator does not. They indicate values of xxx where the function approaches infinity.

Explain the concept of horizontal asymptotes in rational functions.

Horizontal asymptotes describe the behavior of the function as xxx approaches positive or negative infinity. They are determined by comparing the degrees of the numerator and denominator.

Explain the concept of holes in rational functions.

Holes occur when a factor cancels out in both the numerator and denominator. The function is undefined at that xxx-value, but the limit exists.

Explain how to determine the end behavior of a rational function.

Compare the degrees of the numerator and denominator. If the degree of the denominator is greater, y=0y=0y=0. If the degrees are equal, yyy is the ratio of leading coefficients.

Explain the concept of rate of change for linear functions.

The rate of change (slope) is constant for linear functions, meaning the function increases or decreases at a steady rate.

Explain the concept of rate of change for quadratic functions.

The rate of change varies for quadratic functions, meaning the function's increase or decrease is not constant.