4. Polynomial and Rational Functions Flashcards
The quadratic function f(x) = a(x – h)² + k, a ≠ 0, is in ______ form.
The graph of f is called a/an ______ whose vertex is the point _______.
The graph opens upward if a ______ and opens downwards if a _______.
standard
parabola
(h, k)
> 0
< 0
Consider the quadratic function f(x) = ax² + bx + c, a ≠ 0.
If a > 0, then f has a minimum that occurs at x = ______. This maximum value is _______.
if a < 0, then f has a maximum that occurs at x = _____. This maximum value is ______.
– b / 2a
f(– b / 2a)
– b / 2a
f(– b / 2a)
True or False
The graph of f(x) = (x – 2)² + 1 opens upward.
True
True or False
The graph of f(x) = (x + 5)² + 3 has its vertex (5, 3).
False
True or False
The y-coordinate of the vertex of f(x) = 4x² – 16x + 300 is f(2).
True
The difference between two numbers is 8. If one number is represented by x, the other number can be expressed as _______.
The product of the numbers, P(x), expressed in the form P(x) = ax² + bx + c, is P(x) = ______.
x – 8
x² – 8x
The perimeter of a rectangle is 80 feet. If the length of the rectangle is represented by x, its width can be expressed as ______. The area of the rectangle, A(x), expressed in the form A(x) = ax² + bx + c, is A(x) = ______.
40 – x
–x² + 40x
The degree of the polynomial function f(x) = –2x^3(x –1)(x + 5) is _____. The leading coefficient is _____.
5
–2
True or false:
Some polynomial functions of degree 2 or higher have breaks in their graphs.
False
The behavior of the graph of a polynomial function to the far left or the far right is called its _____ behavior, which depends upon the ______ term.
end
leading
The graph of f(x) = x^3 _____ to the left and ______ to the right.
falls
rises
The graph of f(x) = –x^3 _____ to the left and ______ to the right.
rises
falls
The graph of f(x) = x² _____ to the left and ______ to the right.
rises
rises
The graph of f(x) = –x² ______ to the left and ______ to the right.
falls
falls
True or false:
Odd-degree polynomial functions have graphs with opposite behavior at each end.
true
True or false:
Even-degree polynomial functions have graphs with the same behavior at each end.
true
Every real zero of a polynomial function appears as a/an ______ of the graph.
x-intercept
If r is a zero of even multiplicity, then the graph touches the x-axis and _____ at r. If r is a zero of odd multiplicity, then the graph ______ the x-axis at r.
turns around
crosses
If f is a polynomial function and f(a) and f(b) have opposite signs, then there must be at least one value of c between a and b for which f(c) = ______. This result is called the ______ Theorem.
0
Intermediate Value
If f is a polynomial function of degree n, then the graph of f has at most ______ turning points.
n – 1
Consider the following long division problem:
x + 4⟌(6x – 4 + 2x^3)
We begin the division process by rewriting the dividend as ______.
2x^3 + 0x² + 6x – 4
Consider the following long division problem:
3x – 1⟌(6x^3 + 7x² + 12x –5)
We begin the division process by dividing ______ by ______.
We obtain _____.
We write this result above _____ in the dividend.
6x^3
3x
2x²
7x²
In the following long division problem, the first step has been completed:
2x² 5x – 2⟌(10x^3 + 6x² – 9x + 10)
The next step is to multiply _____ and _____.
We obtain _____.
We write this result below ______.
2x²
5x – 2
10x^3 – 4x²
10x^3 + 6x²
In the following long division problem, the first two steps have been completed: 2x 3x – 5⟌(6x² + 8x – 4) 6x² – 10x --------------
The next step is to subtract _____ from ______.
We obtain _____.
Then we bring down _____ and form the new dividend.
6x² – 10x
6x² + 8x
18x
–4