All Flashcards
Explain the concept of end behavior for polynomial functions.
End behavior describes what happens to the function's values as 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 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 is a zero of a polynomial, then 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 where the function approaches infinity.
Explain the concept of horizontal asymptotes in rational functions.
Horizontal asymptotes describe the behavior of the function as 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 -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, . If the degrees are equal, 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 is the general form of a polynomial function?
What is the general form of a rational function?
, where and are polynomials.
How to find the average rate of change of a function over the interval ?
What is the formula for slope (rate of change) of a linear function?
How do you determine the horizontal asymptote of a rational function when the degree of the numerator and denominator are the same?
If , then
How do you determine the horizontal asymptote of a rational function when the degree of the numerator is less than the degree of the denominator?
How do you determine the horizontal asymptote of a rational function when the degree of the numerator is greater than the degree of the denominator?
There is no horizontal asymptote. Consider long division to find slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator.
How do you find the zeros of a polynomial function?
Set and solve for .
How do you find the vertical asymptotes of a rational function?
Set the denominator and solve for , excluding any values that are also zeros of the numerator.
How do you find the holes of a rational function?
Find common factors in the numerator and denominator. The -value where the factor equals zero is 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 .
What is a vertical asymptote?
A vertical line where the function approaches infinity or negative infinity as approaches .
What is a horizontal asymptote?
A horizontal line that the function approaches as 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 approaches positive or negative infinity.