real data - clayton.k12.mo.us of equations= group of equations that share common characteristics...
TRANSCRIPT
Mikayla Johnson in a group with Jordan Shields and Tayla Slay and Steven Williams
Chapter 8: Root and Radicals and Root Functions
Section 1: Radical expressions and Graphs
Section 2: Rational exponents
Section 3: Simplifying Radical expressions
Section 4: Adding and Subtracting Radical exponents
Section 5: Multiplying and Dividing Radius expression
Section 6: Solving Equation with Radius
Section 7: Complex Numbers
Chapter 1 list of topics:
Section 1.1
Identities and contractions
solving for a specified variable in literal equations
Section 1.2
the additive property of inequality
the multiplicative property of inequalities
solving compound inequalities
Section 1.3
Solving absolute value equations
"Less than" absolute value inequalities
"Greater than" absolute value inequalities
Section 1.4
Indemnifying and simplifying imaginary and complex numbers
Adding and subtracting complex numbers
Division of complex numbers
Section 1.5
Quadratic equations and the zero product property
Quadratic equations and the square root property of equality
Solving quadratic equations by completing the square
Solving quadratic equations using the quadratic formula
Section 1.6
Polynomial equations of higher degree
Rational equations
Radical equations and equations with rational exponents
Equations in quadratic form
2.1: Rectangular Coordinates;
Graphing Circles,
Other Relation
2.2: Linear
Graphs
Rates of Change
2.3: Graphs
Special Forms of Linear Equations
2.4: Functions,
Function Notation,
the Graph of a Functions
2.5: Analyzing the graph of a function
2.6: Linear Functions
Real Data
Chapter 3
3.1 The toolbox function
transformation
3.2 Basic rational functions
power function
3.3 Variation:
The toolbox function
power function
3.4 PiecewiseDefined function
3.5 The algebra
Composition of functions
3.6 Another look at formulas,
functions,
problem solving
Section 1
Radian: Is the “a” √n a
Index: Is n √n a
Radical: Is the expression
Principal Root: is the notation √n a
Negative nTh Root:√n a
Radical expression: Algebraic expression that contains radicals
Square root function:
Cube root function:
Section 3
Simplified: Has no factor raised to a power greater than or equal to index, radicand has no
fraction, no denominator contains radical
Pythagorean theorem: Relates the lengths of the tree sides or a right triangle
Hypotenuse: Longest side
Legs: Two Shorter sides
Section 4
Distance formula: d =√(x ) y − )2 − x12 + ( 2 y1
2
The root of the sum does not equal the sum of the root: =√9 6+ 1 / √9 + √16
Section 5
Conjugates: x=y and xy are the congregates, example: is 11 + √2 − √2
Radical equation: An equation that includes one or more radical expressions with a variable
Section 6
Power rule: Raising both sides to a power
Extraneous solutions: Solutes that do not satisfy the original equation
Isolate radical: Make sure that one radical term is alone on one side of the equation
Apply power rule: Raise both sides of the equation to a power that is the same as the index of the
radical
Check: All proposed solutions in the original equation
Character independence: z in z = √l ÷ c
Section 7
Imaginary unit : Defined as i and i −i = √− 1 2 = 1
Complex number: ia = b
Real part: is a of ia = b
Imaginary part: b of ia = b
Pure Imaginary Number: and b =a = 0 / 0
Chapter 1
Section 1
Families of equations= group of equations that share common characteristics
Equation= a statement that 2 expressions are equal
Solutions/roots= replacement values that make the equation true
Equivalent equations= a sequence of simpler equations one simpler than the next next until it
reaches an obvious solution
Backsolution= checking a solution by plugging the solution back into the original problem
LCD= least common denominator
Contradiction equation= an equation that is true for one value, but false for another
Identity= equation that is always true
Contradictions= equations that are never true
Literal equation Do= one with two or more variables
Formula= models a known relationshi
Object variable= the variable that is being solved for
Section 1.2
Solution set= set of numbers that satisfy an inequality
Interval notation= symbolic way of indicating a selected interval of real numbers
Compound inequalities= applications of inequalities with more than one solution interval
Intersection= intersection of two sets A and B written A B set of elements common on both⋂
sides
Union= A B set of elements that are in either set (or both)⋃
Joint inequality= when the original inequality can be joined with another
Disjoint (disconnected) intervals= excluding a number or numbers
Non complex number: ia = b
Complex conjugates: i and a ia = b − b
Exact/closed formula= answers written using radicals
Quadratic formula= general solution used to solve equations belonging to the quadratic family
Definitions: For chapter 2
Section 1
Relation: Correspondence between two sets
Mapping Notation: Showing multiple corresponding values to another value
Dependent Variable: The value that changes due to something
Independent Variable: The value that changes by itself
Domain: Set of all first coordinates
Range: Set of all second coordinates
Equation form: Another name for relations
Tangular Coordinate System: Showing a relation on a graph
Xaxis: Horizontal lines on a graph
Yaxis: Vertical lines on a graph
XYplane: Flat two dimensional surface
Quadrants: Graph divided into four regions
Grid Lines: Shown to denote integer values on a graph
Coordinate Grid: Where the grid lines divide into
Continuous: Graphs that go beyond forever
Parabola: When x or y is squared
Vertex: Lowest point on a graph
Vertical Parabola: Parabola stretching upwards
Semicircle: Half of a circle in two different quadrants
Midpoint: Center between two endpoints
Average Distance: Average between two endpoints
Distance Formula: Find the distance between two points
Radius: Fixed distances
Center: Fixed point
Central Circle: When both x and y coordinates of a circle are 0
General Form: Basic form for an equation
Section 2
YIntercept: When X is zero and the Y has a value
XIntercept: When Y is zero and the X has a value
Intercept Method: Graphing a linear equation using two points.
Slope of a Line: Rate of a line
Rate of Change: One quantity compared to another quantity in measurement
Delta: Represents change
Parallel Lines: Two lines in a plane that do not intersect
Perpendicular Lines: Two lines in a plane that intersect at right angles
Section 3
Secant Line: Straight line through two points on a non linear graph
PointSlope Form: Isolating the Y value in a equation
Section 4
Function: Relation where each element of the domain corresponds to exactly one element of the
range
Vertex: The lowest value of the graph
Vertical Line Test: When a function has each X value on a vertical line once
Point of Inflection: Where the pivit point curves on a graph
Implied Domain: Set of all real numbers for which the function represents a real number
Function Notation: Used to express functions such as f(x)
Section 5
Even Functions: Symmetric about the Yaxis
Maximum Values: YCoordinate peaks from other graphs
Global Maximum: Largest yvalue over the entire domain
Local Maximum: Largest range value in a specified interval
Endpoint Maximum: Endpoint of the domain
Odd Functions: Symmetric about the origin
EndBehavior: Describes the graph as it increases
Average Rate of Change: Slope of a secant line
Section 6
Scatterplot: Graph of all the order pairs in a data set
Positive Association: When data has larger input and output values
Negative Association: When data decreases left to right
Chapter 3
3.1: The Toolbox Functions and Transformations
The id function: f(x)= x
Square root function: f(x)= √x
Cubing function: f(x)= x^3
Squaring function: f(x)= x^2
Absolute value function: f(x)= x| |
Cube root: f(x)= √3 x
Toolbox Functionsgives us a variety of “tools” to model the real world.
Function Family set of functions defined by the same type of formula
Parent Function simplest function of a family of functions
Transformation when the parent graph may become “morphed” and/or shifted from
its original position, yet the graph will still retain its basic shape and features.
Axis of Symmetry a line of symmetry for a graph.
Vertical Shift/ Translation a translation in which the graph is moved up or down on the
yaxis
Horizontal Shift/ Translation a translation in which the graph is moved left or right on
the xaxis
Vertical Reflection translation in which the graph is reflected over the xaxis
Horizontal Reflection transformation in which the graph is reflected over the yaxis
Vertical Stretches transformation in which the graph is stretched due to the lead
coefficient being larger than one
Compression a transformation in which the graph is compressed due to the lead
coefficient being smaller than one
3.2: Basic Rational Function and Power Function
A rational function is one of the form V(x)= (where p and d are polynomials)d(x)p(x)
Rational function the ratio of two polynomials
Reciprocal square function are both above the xaxis
Asymptotic behavior becoming increasingly exact as a variable approaches a limit,
usually infinity
Horizontal asymptotic The line y=0 (the xaxis)
Vertical asymptote the line x=0 (the y=axis)
Power function where x is raised to some power
Allometric studies one area where power function and modeling with regression are
used extensively
3.3; Variation: The Toolbox Functions in Action
Direct Variation when the yvalue varies directly with the xvalue (as x increases, y
increases) y=kx
Constant of Variation in a direct variation is the constant (unchanged) ratio of two
variable quantities
Inverse Variation when the yvalue varies indirectly with the xvalue (as x increases, y
decreases) y=( )x1
Combined Variation when an equation involves both direct, and indirect variation.
Join Variation when one quantity is directly proportional to several others
Suppose a varies directly with b and inversely with c, find a when b=10 and c=5
The time it takes for a simple pendulum to complete one period (swing over and back)
is directly proportional to the square root of its length. If a pendulum 6 ft long has a
period of 3 seconds, find the time it takes for a 18ft long pendulum to complete one
period.
A car’s speed varies directly with the radius of its tires and output (in rpm) of its
engine, but inversely with the product of the transmission and differential gear ratios.
If a car has a top speed of 50 mph at 5000 rpm in top gear in top gear(transmission
ratio of 1.0) what is its speed in first gear (transmission ratio of 2.5)
3.4: PiecewiseDefined Functions
Piecewisedefined function graphs may be various combinations of smooth/not
smooth and continuous/not continuous
Step function the pieces of the function form a series of horizontal step
Ceiling function is the largest integer not greater than x
Floor Function is the smallest integer not less than x
3.5: The Algebra and Composition of Functions
The notation used to represent the basic operation on two function is
(f + g)(x)= f(x) + g(x)
(f g)(x)= f(x) g(x)
(f g)(x)= f(x) g(x)∙ ∙
( )(x)= ;g(x)fg
f(x)g(x) =/ 0
Algebra of Functions basic operations of functions
Composition of Functions when the input value is itself a function (rather than a single
number or variable)
Chapter 8 Equations
Section 1
Basic radical expression: (x) and f(x)f = √x = √3 x
Section 2
If is a real number then √n a a1/n = √n a
If m and n are positive integers with m/n in lowest terms then a )a−m/n = ( 1/n m
If (a = )am/n = 1am/n / 0
If all indicated roots are real numbers then a ) a )am/n = ( 1/n m = ( m 1/n
ar * as = ar+s a ) −r = ( 1ar as
ar = ar−s ( ) ba −r = ar
br (a ) r s = ars (ab) b r = ar r ( ) ba r = br
ar
)a−r = (a1 r
Section 3
If are real numbers an n is a natural number then and √n a √n b √n a * √n b = √n ab
ab)√n a * √n b = a1/n * b
1/n = ( 1/n = √n ab
If are real numbers and n is natural number then and √n a √n b =b / 0 √n ba * √n a
√n a
If m is an integer, n and k are natural numbers, and all indicated roots exist then √kn akm = √n am
c2 = a2 + b2
a= x|| 2 − x1||
Distance between and x , )( 2 y2 x , )( 2 y1
b = y|| 2 − y1||
From pythagrium theorem d2 = a2 + b2 d x ) y ) 2 = ( 2 − x12 + ( 2 − y1
Section 6
Radical equation x )( + y 2 = x2 + 2xy + y2
Section 7
√− b = i√b
Section formulas and equations for chapter 1:
Section 1.4
Standard form = (a+bi)
Complex conjugates= (a+bi) (abi)
Non complex number: ia = b
Complex conjugates: i and a ia = b − b
Section 1.5
Quadratic equations= a +bx+cx2
Square root property= = Kx2
Discriminant= 4ac= determines the nature (real or imaginary) and the number of solutions tob2
a given quadratic equation
Chapter 2 Section 1
Midpoint: ,( 2x₁+x₂
2y₁+y₂)
Distance Formula: ² √(x₂ ₁)− x +√(y₂ ₁)− y
Equation of a Circle: ²+(x )− h ²(y )− k ² = r
h=xcoordinate, k=ycoordinate, r=radius
Section 2
Linear Equations: ax+by=c
a, b, and c are real numbers
Slope Formula: m= x₂−x₁y₂−y₁
Horizontal Lines: y=k (0,k) is the yintercept
Vertical Lines: x=h (h,0) is the intercept
Section 3
Slope Intercept Form: y=mx+b
m is the slope
yintercept is (0,b)
Point Slope Form: yy₁=m(xx₁)
m is the slope
Section 5
Even Functions: f(x)=f(x)
Odd Functions: f(x)=f(x)
Average Rate of Change: xy
Problems
C0S4nc 1. Simplify
6√36 + 6
3√16 + 3√36
C0S7nc 2. Symplify
) − )(√− 6 + i2 + ( i2 + 3
C0S3nc 3. Symplify
43/12 * 48/12
C0S5nc 4. Find Z
h = √ z4l
C0S6nc 5. Simplify
√106592
CHAPTER 1:
All calculator problems. Simplify to the fullest, leave in exact form.
1.) 7x+20=18
2.) √x 5− 3 =
3.) 3x25 2≺ 4
4.) √− 26
5.) 3x x 2 − 7 − 6 = 0
Chapter 3
1. (C3S1nc) Identify the function family
a. f(x)= 3(x+2)^2 5
b. f(x)= 21 x| + 4|
c. f(x)= √3 x − 4 + 2
d. f(x)= √3 x 6+ 1
2. (C3S1nc) Explain how the listed equations affect the function f(x)= (x)^2
a. f(x)= 3(x10)^2
b. f(x)= (x+9)^2 361
c. f(x)= 2(x+4)^2 + 6
3. (C3S1nc) Name The Parent function
Sketch the graph of each function, state transformations, label horizontal and vertical
asymptotes, and x and y intercepts, state domain and range
1. f(x)= +6x1
2. f(x)= 1x−6
3. f(x)= 2x−1
4. f(x)= 8−1(x−2)2
Answer for chapter 1 questions
1. .53
2. i√6 + 3
3. 411/12
4. z = 4lh2
5. 0, 591 6
1.(C3S3nc) Graph and name range and domain
2.(C3S4c) Choose one of the following graphs from the question above, and evaluate the
piecewisedefined function f(3), f(4), and f(6)
(C3S5nc) For f(x)= x^2 +x and g(x) = 3x2, find the following:
a. (f+g)(x)
b. (fg)(2)
c. (f g)(3)∙