Explain the concept of end behavior for polynomial functions.
End behavior describes what happens to the function's values as x approaches positive or negative infinity. It's determined by the leading term (degree and leading coefficient).
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Explain the concept of end behavior for polynomial functions.
End behavior describes what happens to the function's values as x 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 x 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=a is a zero of a polynomial, then (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 x where the function approaches infinity.
Explain the concept of horizontal asymptotes in rational functions.
Horizontal asymptotes describe the behavior of the function as x 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 x-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=0. If the degrees are equal, y 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.
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.
How do you find the zeros of a polynomial function?
Set the function equal to zero, factor the polynomial (if possible), and solve for x. Use the quadratic formula if it's a quadratic.
How do you find the vertical asymptotes of a rational function?
Factor the numerator and denominator. Simplify the rational function. Set the denominator equal to zero and solve for x.
How do you find the horizontal asymptote of a rational function?
Compare the degrees of the numerator and denominator. If numerator degree < denominator degree, y=0. If degrees are equal, y= ratio of leading coefficients. If numerator degree > denominator degree, there is no horizontal asymptote.
How do you identify holes in a rational function?
Factor the numerator and denominator. Identify common factors that cancel out. The x-value that makes the canceled factor zero is the location of the hole.
How do you determine the end behavior of a polynomial function?
Identify the degree and leading coefficient. If the degree is even and the leading coefficient is positive, both ends go up. If the degree is even and the leading coefficient is negative, both ends go down. If the degree is odd and the leading coefficient is positive, the left end goes down, and the right end goes up. If the degree is odd and the leading coefficient is negative, the left end goes up, and the right end goes down.
How do you sketch the graph of a rational function?
Find zeros, vertical asymptotes, horizontal asymptote, and holes. Plot these features on a coordinate plane. Determine the sign of the function in each interval created by the zeros and vertical asymptotes. Sketch the graph based on this information.
How do you find the average rate of change of a function over an interval?
Calculate the function's value at the endpoints of the interval, f(a) and f(b). Use the formula: b−af(b)−f(a).
How do you determine if a function has complex zeros?
If the polynomial has real coefficients and the discriminant (b2−4ac) of a quadratic factor is negative, then the function has complex zeros.
How do you use the Factor Theorem to find zeros?
If f(a)=0, then (x−a) is a factor of f(x). Use synthetic division to divide the polynomial by (x−a) and find the remaining factors.
How to solve for zeros in a polynomial using long division?
If you know one factor of the polynomial, divide the polynomial by that factor using long division. The quotient will be a polynomial of lower degree, which may be easier to factor to find the remaining zeros.