Algebra and Trigonometry: Polynomial and Rational Functions flashcards

Question 1

What is a quadratic function?

Accepted Answer

A polynomial function of degree two.

Question 2

What is the graph of a quadratic function called?

Accepted Answer

A parabola.

Question 3

What is the axis of symmetry of the parabola f(x) = ax^2 + bx + c?

Accepted Answer

The vertical line x = −b/(2a), through the vertex.

Question 4

What is a power function?

Accepted Answer

A function f(x) = kx^p, with constants k and p and a variable base.

Question 5

What is the degree of a polynomial?

Accepted Answer

The highest power of the variable in it.

Question 6

What is the leading coefficient of a polynomial?

Accepted Answer

The coefficient of the term with the highest power of the variable.

Question 7

What determines the end behavior of a polynomial function?

Accepted Answer

Its leading term.

Question 8

At most how many turning points can a polynomial of degree n have?

Accepted Answer

n − 1

Question 9

At most how many x-intercepts can a polynomial of degree n have?

Accepted Answer

n

Question 10

What is a turning point?

Accepted Answer

A point where the graph of a function changes direction.

Question 11

What is the multiplicity of a zero x = h?

Accepted Answer

The power p of the factor (x − h)^p in the factored polynomial.

Question 12

What does a polynomial graph do at a zero of odd multiplicity?

Accepted Answer

It crosses the x-axis.

Question 13

What does a polynomial graph do at a zero of even multiplicity?

Accepted Answer

It touches the x-axis.

Question 14

What does the Intermediate Value Theorem say for a polynomial f with f(a), f(b) of opposite signs?

Accepted Answer

At least one c between a and b has f(c) = 0.

Question 15

Division Algorithm: f(x) divided by a nonzero d(x) with deg d ≤ deg f gives what?

Accepted Answer

Unique q(x) and r(x) with f(x) = d(x)q(x) + r(x), where r(x) = 0 or deg r < deg d.

Question 16

What does the Remainder Theorem state?

Accepted Answer

Dividing a polynomial f(x) by x − k leaves remainder f(k).

Question 17

What does the Factor Theorem state?

Accepted Answer

f(k) = 0 exactly when (x − k) divides f(x) as a factor.

Question 18

Under the Rational Zero Theorem, what form do possible rational zeros p/q take?

Accepted Answer

p divides the constant term and q divides the leading coefficient.

Question 19

What does the Fundamental Theorem of Algebra state?

Accepted Answer

A polynomial of positive degree always has a complex zero (at least one).

Question 20

What does Descartes' Rule of Signs determine?

Accepted Answer

Upper bounds on how many positive and negative real zeros f has, using sign changes in f(x) and f(−x).

Question 21

What is a rational function?

Accepted Answer

A function expressible as one polynomial divided by another.

Question 22

Where does a rational function have vertical asymptotes?

Accepted Answer

Where the denominator is zero and the numerator is not zero.

Question 23

What is inverse variation?

Accepted Answer

A relationship between two variables whose product is constant.

Algebra and Trigonometry: Polynomial and Rational Functions flashcards

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