Quadratic Functions
Rewrite the quadratic function in standard form by completing the square.
f(x)=x2+6x-3
(4.1 #23)
f(x) = (x+3)2-12
Determine if the function is a polynomial. If so, state its degree.
f(x)= x1/3+7x2-2
(4.2 #7)
Not a polynomial
Divide using long division
(x2-5x+6) / (x-2)
(4.3 #3)
x-3 r0
Determine whether the number given is a zero of the polynomial
-3, P(x)= x3 - x2 - 8x + 12
(4.4 #9)
Yes, the given number is a zero of the polynomial
Determine the domain of the rational function
f(x) = 1/ (x+3)
(4.6 #1)
(-infinity, -3) U (-3, infinity)
Find the vertex of the parabola associated with the quadratic function.
f(x)=332 -2x+15
(4.1 #41)
(1/33 , 494/33 )
Find all the real zeros and state their multiplicities of the polynomial function.
f(x)= 4x2(x-1)2(x2+4)
(4.2 #31)
zero at 0,multiplicity of 2;zero at 1,multiplicity of 2
Divide using long division
(9x2-25) / (3x-5)
(4.3 #11)
3x+5 r0
Given a real zero of the polynomial, determine all other real zeros, and write the polynomial in terms of a product of linear and/or irreducible quadratic factors.
Polynomial:
P(x) = x3- 13x +12
Zero:
1
(4.4 #11)
1,3, -4; P(x)= (x-1)(x-3)(x+4)
Find all of the vertical asymptotes and horizontal asymptotes
f(x) = (6x2 + 4x +5) / (6x2 + 5x - 4)
(4.4 #15)
HA: none
VA: x = 1/2 and x = -4/3
Find the quadratic function that has the given vertex and goes through the given point.
vertex: (2,5) point: (0,1)
(4.1 #51)
y = -5(x-2)2 +5
Find a polynomial of minimum degree that has the given zeros (there are multiple answers)
-5, -3, 0 ,2, 6
(4.2 #41)
P(x) = x(x+5)(x+3)(x-2)(x-6)
Divide using synthetic division
(7x2-3x+5) / (x+1)
(4.3 #33)
7x-10 r15
During the course of treatment of an illness the concentration of a dose of a drug (in mcg/mL) in the bloodstream fluctuates according to the model
C(t) = 15.4 - 0.05t2
where t = 0 is when the drug was administered. Assuming a single dose of the drug is administered, in how many hours (to the nearest hour) after being administered will the drug be eliminated from the bloodstream?
(4.4 #77)
18 hours
True or false?
A rational function can have either a horizontal asymptote or a slant asymptote, but not both
(4.6 #79)
True
WORD PROBLEM!
The number of cell phones in the United States can be approximated by a quadratic function. In 1996 there were approximately 16 million cell phones, and in 2005 there were approximately 100 million. Let t be the number of years since 1996. The number of cell phones in 1996 is represented by (0, 16), and the number in 2005 is (9, 100). Let (0, 16) be the vertex.
Based on this model, how many cell phones were used in 2010?
(4.1 #75 B)
219 million cell phones
Determine the end behavior of the polynomial function
f(x)= x(x - 2)3.
Solution:
This polynomial has similar end behavior to the graph of y = x3.
End behavior falls to the left and rises to the right. This is incorrect. What mistake was made?
(4.2 #88)
the power will be 4, making the end behavior similar to a quadratic
The area of a rectangle is 6x4+4x3-x2-2x-1 square feet. If the length of the rectangle is 2x2-1 feet, what is the width of the rectangle?
(4.3 #61)
The width is 3x2+2x+1 feet
Use Descartes’ rule of signs to determine the possible number of positive real zeros and negative real zeros.
P(x) = x4 - 32
(4.4 #33)
negative real zeros = 1
Determine the asymptotes of the rational function
f(x)= (x-a)(x+b) / (x-c)(x+6)
(4.6 #83)
HA: y=1
VA: x=c and x= -d
State whether the following statements are true or false:
1. A quadratic function must have a y-intercept
2. A quadratic function may have more than one y-intercept
(4.1 #83, 85)
1. true
2. false
For the polynomial, list each real zero and its multiplicity, and determine whether the graph touches or crosses at each x-intercept
f(x)= 2x5-6x4-8x3
(4.2 #65)
Zeros at -1, 0 and 4; has a multiplicity 3, -1 and 4 each have multiplicity of 1
The graph crosses at all of the zeros
Is x+b a factor of x3 + (2b - a)x2 + (b2 - 2ab)x- ab2?
(4.3 #73)
Yes
True or false?
A polynomial of degree n, n > 0, can be written as a product of n linear factors over real number
(4.4 #83)
False
A 500-square-foot rectangular garden will be enclosed with fencing. Write a rational function that describes how many linear feet of fence will be needed to enclose the garden as a function of the width of the garden w.
(4.6 #69)
(2w2+1000) / w