Write the long-run behavior of f(x)=x^6 using arrow notation
As x-->inf, f(x)-->inf
As x-->-inf, f(x)-->inf
100
Find the value that completes the square and then re-write as a perfect square:
y^2 - y + ____
1/4; (y-1/2)^2
100
What is the quadratic formula?
x = -b +- sqrt(b^2 - 4ac)
2a
100
Use intercepts to sketch the graph of
f(x)=x^2-2x
parabola opens up with x-int at 0 and 2, y-int at 0
200
Factor: 4uv + 14u^2 + 12v + 42u
2(u+3)(2v+7u)
200
If a graph has 3 x-intercepts, 4 extrema, and a negative leading coefficient, what is the least possible degree of the polynomial? Sketch a graph that fits these criteria.
degree 5 (nth degree for n-1 extrema)
graphs may vary
200
Find the vertex of
f(x)=5x^2+2x-2
(-1/5,-11/5)
200
Solve the following quadratic equation by factoring:
5x^2 - 13x = 28
x = -1
x = -5
200
Graph f(x)=2x^2+12x+10
Parabola opens up
vertex is at (-3,-8)
y-int is at (0,10)
300
Factor:
49x^2 + 14x +1
(7x+1)^2
300
What is a polynomial? Define in your own words. Describe the function and the graph.
A sum of terms where the powers on the variable are all positive integers. The graph can be drawn with smooth curves without lifting a pencil, so there are no sharp corners or gaps. Standard form of the equation: f(x)=a+bx+cx^2+dx^3+...+ex^n
300
Find the value that completes the square, then write it as a perfect square
x^2 - 5/4x + ___
25/64; (x-5/8)^2
300
Use the Zero Product Property to solve the following quadratic equation.
15x^2 - 29x -14 = 0
(x-4)(5x+7)=0
x = 4
x = -7/5
300
Myla has $80 to spend on a fence for her rectangular garden. She wants to use cedar fencing which costs $8/meter on one side, and cheaper metal fencing which costs $2/meter for the other three sides. What is the largest area she can enclose, and what are the dimensions (length and width)?
Length of cedar side = 4 meters
Width of metal side = 10 meters
Largest Area = 40 m^2
400
Factor:
3x^5 - 48xy^4
3x(x^2+4y^2)(x+2y)(x-2y)
400
Given f(x)= -3x^3 + 6x^2 + 24x, find the intercepts and long-run behavior. Then graph the polynomial
x-int: (0,0), (-2,0), (4,0)
y-int: (0,0)
As x-->inf, f(x)-->-inf
As x-->-inf, f(x)-->+inf
400
Use completing the square to re-write in vertex form:
f(x)=-2x^2-4x-5
f(x)=-2(x+1)^2-3
400
Solve using the quadratic formula:
2x^2 + 9x + 4 = 0
x = -1/2
x = -4
400
A farmer has 500 feet of fencing to enclose a rectangular pen next to a barn. He only needs fencing for 3 sides because one of the sides is the barn. What are the dimensions (length and width) that would maximize the area?
A(x)=w(500-2w)
w=125 ft
A=31,250 ft^2
L = 250 ft
500
Factor:
125x^3 - 216
(5x-6)(25x^2+30x+36)
500
Write an equation for a polynomial that is degree 5, roots of multiplicity 2 at x=3 and x=1, root of multiplicity 1 at x=-3, and y-intercept at (0,9). Then graph the polynomial.
f(x)=1/3(x-1)^2*(x-3)^2*(x+3)
500
Use completing the square to re-write in vertex form:
f(x)= -4x^2 +32x-67
f(x)= -4(x-4)^2-3
500
Solve using any method:
x^2 - 6x + 9 = 0
x = 3 +/- sqrt(2)
500
A ball thrown in the air by a child can be modeled by
h(x)= -1/14x^2 + 4x + 5, where both the vertical and horizontal distance are measured in feet.
a) How high is the ball when it leaves the child's hand?
b) What is the maximum height of the ball?
c) How far from the child does the ball strike the ground?
a) This is the y-int: h(0)=5 feet
b) This is the k-value of the vertex, k=f(-b/2a) = 61 feet
c) This is the positive x-int: x=57.223 feet (use quadr formula).