Math B: Ms
Math B: Ms. Varuzza
What You Need to Know For the Math B Regent Exam!!!
Circle Geometry:
Characteristics (facts)
- has a total of 360º or 2[pic] radians
- a diameter passes through the center of the circle cutting the it into two semicircles each contain 180º or radians
- tangent is a segment outside a circle and touches the circle once
- secant is a segment that passes through the circle and intersects it twice
- chord is a segment contained inside the circle
- central angle is an angle whose vertex is the center of the circle
- inscribed angle is an angle whose vertex in on the circle
- arc is a portion of a circle intercepted by an angle
Formulas
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|central angle = measure of intercepting arc |inscribed angle = ½ of intercepting arc |
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|angles formed by two cords inside circle = |angle outside of circle = big arc – small arc |
|arc + opposite arc |2 |
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|tangent and a secant |two secants |
|tangent2 = (secant)(external secant) |(secant)(external secant) = (secant)(external secant) |
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|two cords intersecting within the circle |Angle formed by a tangent and cord = ½ of the arc it cuts off |
|(segment1)(segment2) = (segment3)(segment 4) | |
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|Area of Circle = r2[pic] |Circumference of Circle = [pic]d or 2[pic]r |
Quadratics: ax2 + bx + c = 0
1. forms a parabola
-x2 ( downward parabola; frown + x2 ( upward parabola; smile
-to find roots: [pic]
- sum of roots [pic] -product of roots [pic] -axis of symmetry/vertex x = [pic]
2. nature of roots: If b2 – 4ac =
- 0 the roots are real, rational and equal; there is one root
- perfect square root, the roots are real, rational and unequal; there are two roots and touches axis twice
- whole number, roots are real, irrational and unequal; two roots and touches axis twice
- negative, roots are imaginary, no real roots; does not touch the x-axis
Equations of Graphs:
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|Equations |Graph |
|y = mx + b |Linear Equation (Straight Line) |
|Y = ax2 + bx + c |Parabola (U-Shaped) |
|ax2 + by2 = r2 |Circle (a = b) |
|ax2 + by2 = c2 |Ellipse (a ≠ b) |
|ax2 – by2 = c2 and/or y = [pic] |Hyperbola |
|y = #x |Exponential Function |
Radians: (360º = 2[pic] radians)
- change degrees to radians, multiply by [pic]
- change radians to degrees, multiply by [pic]
- to find the intercepting arc given the radius and central angle Ө = [pic] (Ө = central angle, s = intercepting arc and r = radius)
Transformations:
1. reflections of 2. rotations
- x-axis ( negate y - positive: counterclockwise
- y-axis ( negate x - 90º ( negate y and switch
- origin ( negate both - 180º ( negate both
- y =x ( switch x and y - 270º ( negate x and switch
- y = -x ( switch and negate both x and y -negative: clockwise
- -90º ( same as 270º
- -180º ( same as 180º
- -270º ( same as 90º
3. translations 4. dilation
- add to move up to right - multiple same number to x and y to make bigger
- subtract to move down or left - divide same number to x and y to make smaller (or
multiply by a fraction
| |Reflection |Rotation |Dilation |Translation |
|Isometry – distance doesn’t |Yes |Yes |Yes |No |
|change | | | | |
|Direct Isometry – orientation |No |Yes |Yes |Yes |
|stays the same | | | | |
|Opposite Isometry – orientation |Yes |No |No |No |
|changes | | | | |
remember: If a question asks (rx-axis◦ D2), you must do the dilation first and then the reflection.
Imaginary Chart: Square Roots: (always simplify radicals)
i0 = 1 To Add/Subtract: need the same radicands
i1 = i [pic]( [pic]([pic]
i2 = -1 To Multiply/Divide: multiply/divide all like terms
i3 = -i [pic]
Radical Equations:
to solve radical equations, isolate the radical and then raise it to its index (ex: raise a square root to a power of two)
[pic]((isolate radical) [pic]((square both sides) [pic]( (solve the algebraic equation) [pic]
ALWAYS CHECK YOUR SOLUTION BECAUSE IT DOES NOT ALWAYS WORK IN THE EQUATION!
Logarithms:
means/equals EXPONENT rewrite logarithms: [pic] ( [pic]
**REMEMBER, IF LOG HAS A BASE OF 10, THEN YOU CAN PUT IT IN THE CALCULATOR**
**THE WORD LOG WITH NO BASE NUMBER WRITTEN IS UNDERSTOOD AS THE BASE OF 10**
Logarithm Rules: Exponent Rules:
[pic] [pic]
Exponential Equations:
1. need to get the base of the powers to be the same/equal
[pic]((decide what is the common base) [pic]([pic]((when bases are the same, set exponents equal to each other and solve algebraic equation) 3(2x + 1) = 2(4x)
2. when you cannot get the base powers to be the same, need to use logarithms
34x+1 = 52((log both sides)[pic]((use any logarithm rules)[pic]((solve for x by diving log 3, subtracting 1 and dividing 4)
Absolute Values:
the positive value of a number
Solving an Absolute Equation: set equation equal to both the negative and positive solution and solve
[pic] ( [pic] ( [pic]
Solving an Absolute Inequality:
-greater than/greater than or equal to-solve normal equation and solve equation with a negative solution but change sign position (graph is separate)
[pic] ( [pic] ([pic]
-less than/less than or equal to-solve normal equation and solve equation with a negative solution and keep sign position (graph is together)
[pic] ( [pic]
Functions:
has one and only one domain (can have repeating ranges)
domain = x-values range = y-values
to find the domain of a function, find out where on the graph the function is ( explain what are the values of the x-coordinates; what makes the function undefined
remember: if you have a function where division is involved, you cannot divide by 0
[pic] ( solve for x2 – 4 = 0 and x cannot be that value
if the function is a radical, radicals cannot equal negative solutions
[pic]( solve for this equation and x has to be that value and higher
if the function includes both a radical and division, the function cannot equal 0 or anything less.
[pic] ( solve for [pic] and x cannot be that value or lower
Math A Material:
Slope: [pic] Distance: [pic] Midpoint: [pic]
**NOTE: TRIGONOMETRY IS NOT ON THIS SHEET BECAUSE YOU HAVE BEEN GIVEN MANY HANDOUTS WITH INFO FOR IT***
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