Quadratic Equations
Write the following equation in the quadratic formula (you do not need to solve).
2x2 - 6x + 8 = 0
[6 +/- sqrt((-6)2 - 4(2)(8))] / 2(2)
y = 2x - 5 is a _______ equation with a slope of _____.
Linear, 2.
Write sine and cosine as ratios of the sides of a right triangle.
sin(x) = opposite/hypotenuse
cos(x) = adjacent/hypotenuse
We can write linear equations in _______ form (y = mx + b) as well as ________ form (y - y1 = m(x - x1)
slope-intercept
point-slope
Solve:
x2 + 4x - 5 = 0
x = -5
x = 1
y = 3x is a _____ equation, and y = x3 is a ________ equation.
exponential, power
Write tangent, secant, and cosecant as ratios of the sides of a right triangle.
tangent = opposite/adjacent
secant = 1/cos(x) = hypotenuse/adjacent
cosecant = 1/sin(x) = hypotenuse/opposite
The process of changing a quadratic equation from standard form to _______ form is called __________.
vertex
completing the square
Complete the Square:
3x2 - 8x + 6 = 0
3(x - 4/3)2 + 2/3
y = (x - 5)2 + 4 is a _______ equation in _______ form.
quadratic, vertex.
What is the domain and range of sin(x) and cos(x)?
(they are the same)
domain: all real numbers
range: [-1, 1]
Explain the distinction between whole numbers and integers.
whole numbers: 0, 1, 2...
integers: ...-2, -1, 0, 1, 2...
Solve, including all solutions
2x2 - 3x + 9 = 0
x = 3/4 +/- [i*3sqrt(7)] / 4
(x + 3)2 + (y - 1)2 = 9 is a ________ equation with a radius of ____, and center at ________.
circle, 3
center = (-3, 1)
What is the Law of Sines and Law of Cosines (write their formulas)
LoS: a/sinA = b/sinB = c/sinC
LoC: c2 = a2 + b2 - 2ab*cosC
What is the distinction between real numbers and rational numbers?
real numbers: all numbers with no imaginary part.
rational numbers: can be written as ratio of integer/integer.
b2 - 4ac is a part of the quadratic formula called the determinant. Explain how solving the determinant gives you the number of real vs complex solutions a quadratic equation will have.
det. positive -> 2 real solutions
det. = 0 -> 1 real solution
det. negative -> 0 real solutions
y = |2x - 3| - 5 is an ________ equation with a vertical shift of ______ and a horizontal shift of ______.
absolute value, VS = -5, HS = 1.5
Explain what the ambiguous case for the Law of Sines means and when it occurs.
Ambiguous Case: 2 possible triangles possible from given information.
Occurs when given SSA, (two sides and a non-included angle).
A manufacturer develops a formula to determine the demand for its product depending on the price in dollars: D = 2000 + 100p - 6p2, where p is the price per unit, and D is the number of units in demand. At what price will the demand drop to 1000 units? Round to the nearest tenth.
~$23.70