Hauptinhalt
Nebraska Math
High School Advanced Topics: ALGEBRA
Students will solve problems and reason with algebra using multiple representations, make connections within math and across disciplines, and communicate their ideas.
AT.A.1
Algebraic Relationships: Students will demonstrate and represent relationships with functions.
Analyze and graph nonlinear functions (trigonometric, rational, higher-order polynomials, logarithmic, and piecewise) and relations (conic sections) using their points of interest and graphing technology.
- Amplitude & period of sinusoidal functions from equation
- Amplitude of sinusoidal functions from equation
- Amplitude of sinusoidal functions from graph
- Discontinuities of rational functions
- Ellipse equation review
- Ellipse foci review
- Ellipse graph from standard equation
- Ellipse standard equation & graph
- Ellipse standard equation from graph
- End behavior of polynomials
- End behavior of polynomials
- End behavior of rational functions
- End behavior of rational functions
- Equation of a hyperbola from features
- Equation of a hyperbola not centered at the origin
- Evaluate piecewise functions
- Evaluate step functions
- Example: Graphing y=-cos(π⋅x)+1.5
- Example: Graphing y=3⋅sin(½⋅x)-2
- Features of sinusoidal functions
- Finding zeros of polynomials
- Foci of a hyperbola from equation
- Foci of a hyperbola from equation
- Foci of an ellipse from equation
- Foci of an ellipse from radii
- Graph & features of ellipses
- Graph of y=sin(x)
- Graph of y=tan(x)
- Graph sinusoidal functions
- Graph sinusoidal functions: phase shift
- Graphing hyperbolas (old example)
- Graphing rational functions according to asymptotes
- Graphs of polynomials
- Graphs of polynomials: Challenge problems
- Graphs of rational functions
- Graphs of rational functions: horizontal asymptote
- Graphs of rational functions: vertical asymptotes
- Graphs of rational functions: y-intercept
- Graphs of rational functions: zeros
- Interpreting trigonometric graphs in context
- Interpreting trigonometric graphs in context
- Intersection points of y=sin(x) and y=cos(x)
- Intro to ellipses
- Intro to end behavior of polynomials
- Intro to hyperbolas
- Introduction to piecewise functions
- Midline of sinusoidal functions from equation
- Midline of sinusoidal functions from graph
- Midline, amplitude, and period review
- Modeling with sinusoidal functions
- Modeling with sinusoidal functions: phase shift
- Multiplicity of zeros of polynomials
- Period of sinusoidal functions from equation
- Period of sinusoidal functions from graph
- Periodicity of algebraic models
- Periodicity of algebraic models
- Piecewise functions graphs
- Positive & negative intervals of polynomials
- Positive and negative intervals of polynomials
- Proof of the hyperbola foci formula
- Rational function points of discontinuity
- Sign of average rate of change of polynomials
- Solving equations graphically: word problems
- Structure in rational expression
- Transforming sinusoidal graphs: vertical & horizontal stretches
- Transforming sinusoidal graphs: vertical stretch & horizontal reflection
- Trig word problem: modeling annual temperature
- Trig word problem: modeling daily temperature
- Trig word problem: solving for temperature
- Vertices & direction of a hyperbola
- Vertices & direction of a hyperbola
- Vertices & direction of a hyperbola (example 2)
- Worked example: domain & range of piecewise linear functions
- Worked example: domain & range of step function
- Worked example: evaluating piecewise functions
- Worked example: graphing piecewise functions
- Zeros of polynomials (multiplicity)
- Zeros of polynomials (multiplicity)
- Zeros of polynomials (with factoring): common factor
- Zeros of polynomials (with factoring): grouping
- Zeros of polynomials & their graphs
- Zeros of polynomials & their graphs
- Zeros of polynomials introduction
- Zeros of polynomials: matching equation to graph
- Zeros of polynomials: matching equation to zeros
- Zeros of polynomials: plotting zeros
Use the unit circle to define the trigonometric functions on multiples of known angles (positive and negative multiples of 30 and 45 degrees or pi/6 and pi/4).
Given a function, list the sequence of algebraic transformations that changes a parent function to the given function.
Define the radian unit of measure and its relationship with degrees.
Explain symmetry of functions and determine whether a function is odd, even, or neither.
- Even & odd functions
- Even & odd polynomials
- Even and odd functions: Equations
- Even and odd functions: Find the mistake
- Even and odd functions: Graphs
- Even and odd functions: Tables
- Function symmetry introduction
- Intro to function symmetry
- Sine & cosine identities: symmetry
- Symmetry of algebraic models
- Symmetry of algebraic models
- Symmetry of polynomials
- Tangent identities: symmetry
Represent, interpret, and analyze inverses of functions algebraically and graphically using domain restrictions when necessary.
- Evaluate inverse functions
- Find inverses of rational functions
- Finding inverse functions: linear
- Finding inverse functions: rational
- Finding inverses of linear functions
- Graphing the inverse of a linear function
- Inputs & outputs of inverse functions
- Intro to inverse functions
- Intro to inverse functions
- Restrict domains of functions to make them invertible
- Restricting domains of functions to make them invertible
Write equations of nonlinear functions (trigonometric, rational, higher-order polynomials, logarithmic and piecewise) using points of interest of the function.
- Construct sinusoidal functions
- Graph sinusoidal functions: phase shift
- Modeling with sinusoidal functions
- Modeling with sinusoidal functions: phase shift
- Positive and negative intervals of polynomials
- Sinusoidal function from graph
- Trig word problem: length of day (phase shift)
- Trig word problem: modeling annual temperature
- Trig word problem: modeling daily temperature
- Zeros of polynomials (multiplicity)
- Zeros of polynomials (multiplicity)
- Zeros of polynomials & their graphs
- Zeros of polynomials & their graphs
- Zeros of polynomials: matching equation to graph
- Zeros of polynomials: matching equation to zeros
Convert between radian and degree measures of an angle.
Use limits to describe the behavior of a function near its asymptotes and removable discontinuities.
- 1-sided vs. 2-sided limits (algebraic)
- Approximating limits using tables
- Conclusions from direct substitution (finding limits)
- Connecting limits and graphical behavior
- Connecting limits and graphical behavior
- Continuity at a point
- Continuity at a point
- Continuity at a point
- Continuity at a point (algebraic)
- Continuity over an interval
- Defined vs. undefined limits
- Estimating limits from tables
- Finding limits by factoring
- Functions continuous at specific x-values
- Functions with same limit at inifinity
- Infinite limits and asymptotes
- Intro to limits
- Introduction to infinite limits
- Introduction to limits at infinity
- Limit of (1-cos(x))/x as x approaches 0
- Limit of piecewise function: limit doesn't exist
- Limit of sin(x)/x as x approaches 0
- Limits at infinity of rational functions
- Limits at infinity of rational functions
- Limits at infinity of rational functions (example)
- Limits at infinity of rational functions: radicals
- Limits at infinity of rational functions: radicals (even power)
- Limits at infinity of rational functions: radicals (odd power)
- Limits at infinity: graphical
- Limits by direct substitution
- Limits by direct substitution
- Limits by factoring
- Limits by rationalizing
- Limits by rationalizing
- Limits from graphs
- Limits from graphs
- Limits from graphs
- Limits intro
- Limits intro
- Limits of combined functions
- Limits of combined functions
- Limits of combined functions: piecewise functions
- Limits of combined functions: sums and differences
- Limits of composite functions
- Limits of composite functions
- Limits of composite functions: external limit doesn't exist
- Limits of composite functions: internal limit doesn't exist
- Limits of piecewise functions
- Limits of piecewise functions
- Limits of trigonometric functions
- Limits of trigonometric functions
- Limits using trig identities
- Next steps after indeterminate form (finding limits)
- One-sided limits from graphs
- One-sided limits from graphs
- One-sided limits from graphs: asymptote
- One-sided limits from tables
- One-sided limits from tables
- Strategy in finding limits
- Strategy in finding limits
- Strategy in finding limits
- Theorem for limits of composite functions: when conditions aren't met
- Trig limit using double angle identity
- Trig limit using pythagorean identity
- Types of discontinuities
- Unbounded limits
- Unbounded limits: algebraic
- Unbounded limits: algebraic (cosine)
- Unbounded limits: algebraic (rational)
- Unbounded limits: graphical
- Undefined limits by direct substitution
- Using tables to approximate limit values
Analyze and model authentic situations using various non-linear representations and relations with appropriate technology.
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Analyze and model authentic application situations using various non-linear representations and relations with appropriate technology.
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