Basics of Algebra
Real Numbers:______
______
Rational Numbers:______
______
ex:
Irrational Numbers:______
ex:
Slope:m =
Equations of Lines:slope – intercept y = mx + b
point – slopey – y1 = m (x – x1)
standardAx + By = C
Parallel:same slope / different “y” intercepts
Perpendicular:slopes are opposite, reciprocals of each other
Find the equation of the line parallel to through the point ( 8, 5 )
Find the equation of the line perpendicular to 4x – 2y = 9 through the point ( 3, -3 )
Functions
Given:f(x) = 2x – 9g(x) = 3x2 + 5x – 1 m(x) = x2 + 4
Evaluate:
f(9)f(2x)g(-7)
g(3x)g(b+2) – g(b – 3)
Domain and Range
Domain:All the input values for a function.
- cannot divide by 0
- cannot take an even root of a negative number
ex.4x3 – 6x + 3ex.
ex.ex.ex.
Range:The output values of a function.
ex.y = 2x + 4ex.y = -2x2+ 4
ex.y = ex.y = ex.
Graphs of piecewise functions:
Composition of functions
Combine functions written in the form f(g(x)) or (f o g) (x)
f(x) = 2x – 3 g(x) = cos x
f(g(x))g(f(x))
f(x) = x2+3, g(x) = x + 1h(x) = f(g(x))
h(3)g(f(-2))g(g(2))
Types of Symmetry
y axis:( -x, y ) and ( x, y ) are mirrored
x and –x yield the same “y” value
ex:y = x2 + 4
= (-x)2+4
=x2 + 4
x axis:( x, y ) and ( x, - y )are mirrored
y and –y yield the same “x” value
ex:x = 4 + y2
= 4 + (-y)2
= 4 + y2
origin:( x, y ) and ( -x, -y )are mirrored
x to –x yield (-) y from original x.
ex:y = 2x3 – x
= 2(-x)3 – (-x)
= -2x3 + x
Even / Odd Functions
Even:reflective about the y axis
ex:y = 1 + cos x
Odd:reflective about the origin
ex:y = x3 – 4xex:y =
Why is there no function reflective over the x axis ?
Transformation Shifts( Hand out WS of families of graphs and fill in first )
f(x) + cShifts the function ______
f(x) – c Shifts the function ______
f(x+c)Shifts the function ______
f(x-c)Shifts the function ______
- f(x)Reflects the function ______
f(-x)Reflects the function ______
ex:- (x – 2 )2 + 1ex:- x3 – 5
ex:ex:cos (-x) – 2
Trigonometry
2π rad = 360° π rad = 180°
Angles
- Positive - ______
- Negative - ______
- Coterminal - ______
______
ex. Calculate 2 positive and 2 negative angles to the following.
-200°
Degrees to Radians to Degrees
Changing Degrees to Radians
30°45°225°40°
Changing Radians to Degrees
Determine Quadrant
Base answer on :
ex:sin x > 0 and cos x > 0ex:cot x < 0 and sec x > 0
Six Trig ratios ( from triangle and from a point )
ex:ex:ex:
Calculus Factoring Worksheet
Factor the following expressions
1) 2)
3)4)
5)6)
7)8)
9) 10)
11)12)
13)14)
Trig – Day 2
Unit Circle
Evaluating special trig values ( no calc )
ɵ = ref : ______
______ ______ ______
______
ɵ = ref : ______
______ ______ ______
ɵ = ref : ______
______ ______ ______
______
ɵ = ref : ______
______ ______ ______
______
ɵ = ref : ______
______ ______ ______
______
ɵ = ref : ______
______ ______ ______
______
Evaluate trig ( calc )
ex:cos ex:csc 40°ex:tan 2.43ex:cot
Trig – day 3
Inverse Trig
sin-1 cot-1 tan
Solving Trig Equations / Identities
sin = sin csc= 1csc =
cos = cossec = 1sec =
tan = tancot = 1cot =
Final all values IB [0, 2π)
ex:2 sin x + 1 = 0ex:tan2x – 1 = 2
cos2x + cos x + sin2x = 06cos2x – cos x – 2 = 0
Trigonometry Worksheet
Evaluate the following:
- cos 3π2.sin π3.sin 4.sec
5.sin 6.csc π7arcsin 8.Arccos
9.arcsin 10.Tan-1 ()11.arcsin 012.arcos 0
Give all six trigonometric ratios for the given angle:
13.r = 14.r = 15.r =
Factoring Review Worksheet
1.2x3 + 10x2 + 12x = 0 2.x2 – 6x + 8= 0
3.2n2 – 13n + 6= 0 4.6x2 – 13x + 6 = 0
5.x3 + 276.8n3 – 125y3
7.(x+1)2 (x+3)3 + (x+1)3 (x+3)28.3(x-3)2(x+9)4 + 4(x-3)3(x+9)3
9.3 (2x-1)2(5x+4)2(2) + 2 (2x-1)3(5x+4)5
Calculus Chapter 1 Review SheetName: ______
- Find the domain and range of: a) b)
- Given , find:a) b) c)
- Identify all types of symmetry in:a) b) c)
- Is the function even, odd or neither:a) b) c)
- Describe the shift:a) b)
- Evaluate the following trigonometric expressions:
a) b) c) d) e)
f) g) h) i)
- Solve the following equations for from :
a) b) c)
- Graph:
- Write the equation of the line that is a) parallel and b) perpendicular to the equation that passes through the point .
- The terminal side of an angle ends at the ordered pair . Find .
- Given and , determine:
a) b) c) d)
- Factor the following:
a) b)
c) d)
- Find a positive and negative coterminal angle to:a) b)
- Convert the following to degrees or radians:a) b)
Answers:
1)a) b)
2)a) 6 b) c) 3) a) None b) Y-axisc) None
4) a) Odd b) Even c) Odd 5) a) Up 4, Right 3 b) Left 1, Reflected over the x-axis
6)a) b) c) d) e) 2 f) g) h) i)
7) a) b) c) 8) See graph below:
9) a) b)
10)
11) a) b) c) 25d) 17
12) a) b)
c) d)
13) a) b)
14) a) b)
1