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Math
Virginia Math
Mathematical Analysis: Characteristics of Functions
Use mathematical reasoning to identify polynomial, rational, piecewise-defined, absolute value, radical, and step functions, given an equation or graph.
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domain and range;
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roots (including complex roots);
- Complex numbers & sum of squares factorization
- Discontinuities of rational functions
- Factor polynomials: complex numbers
- Factoring polynomials using complex numbers
- Finding average rate of change of polynomials
- Multiplicity of zeros of polynomials
- Number of possible real roots of a polynomial
- Positive & negative intervals of polynomials
- Positive & negative intervals of polynomials
- Positive and negative intervals of polynomials
- Quadratics & the Fundamental Theorem of Algebra
- The Fundamental theorem of Algebra
- Zeros of polynomials (factored form)
- Zeros of polynomials (multiplicity)
- Zeros of polynomials (multiplicity)
- Zeros of polynomials (with factoring)
- Zeros of polynomials (with factoring): common factor
- Zeros of polynomials (with factoring): grouping
- 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
intercepts;
- Finding average rate of change of polynomials
- Graphs of polynomials
- Graphs of polynomials: Challenge problems
- Graphs of rational functions
- Graphs of rational functions: y-intercept
- Graphs of rational functions: zeros
- Positive & negative intervals of polynomials
- Positive & negative intervals of polynomials
- Positive and negative intervals of polynomials
- Rational functions: zeros, asymptotes, and undefined points
- Solving equations graphically: word problems
- Zeros of polynomials (factored form)
- Zeros of polynomials (multiplicity)
- Zeros of polynomials (multiplicity)
- Zeros of polynomials (with factoring)
- Zeros of polynomials (with factoring): common factor
- Zeros of polynomials (with factoring): grouping
- 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
symmetry (including even and odd functions);
- Even & odd functions: Equations
- Even and odd functions: Equations
- Even and odd functions: Find the mistake
- Even and odd functions: Graphs
- Even and odd functions: Graphs and tables
- Even and odd functions: Tables
- Function symmetry introduction
- Function symmetry introduction
- Symmetry of algebraic models
- Symmetry of algebraic models
- Symmetry of polynomials
relative and/or absolute maximum and minimum points.
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asymptotes (horizontal, vertical, and oblique/slant;
- Discontinuities of rational functions
- End behavior of rational functions
- Graphing rational functions according to asymptotes
- 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
- Rational functions: zeros, asymptotes, and undefined points
points of discontinuity;
- Classify discontinuities
- Continuity at a point
- Continuity at a point (algebraic)
- Continuity at a point (graphical)
- Discontinuities of rational functions
- Functions continuous at specific x-values
- Graphing rational functions according to asymptotes
- Rational functions: FAQ
- Rational functions: zeros, asymptotes, and undefined points
- Removable discontinuities
- Removing discontinuities (factoring)
- Removing discontinuities (rationalization)
- Types of discontinuities
- Worked example: Continuity at a point (graphical)
intervals for which the function is increasing, decreasing or constant;
Sketch the graph of a polynomial, rational, piecewise-defined, absolute value, radical, and step function.
- Absolute value graphs review
- End behavior of rational functions
- Finding average rate of change of polynomials
- Graph absolute value functions
- Graphing absolute value functions
- Graphing rational functions according to asymptotes
- Graphing square and cube root functions
- 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
- Graphs of square and cube root functions
- Multiplicity of zeros of polynomials
- Positive & negative intervals of polynomials
- Positive & negative intervals of polynomials
- Positive and negative intervals of polynomials
- Radical functions & their graphs
- Shifting absolute value graphs
- Sign of average rate of change of polynomials
Verify estimates about the limit of a function using graphing technology.
- Limits at infinity of quotients with square roots (odd power)
- Limits by factoring
- Limits by factoring
- Limits intro
- Limits intro
- Limits intro
- Limits of combined functions: piecewise functions
- 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
Determine the limit of a function algebraically and verify with graphing technology.
- Functions with same limit at infinity
- Infinite limits and asymptotes
- Infinite limits: algebraic
- Infinite limits: graphical
- Introduction to infinite limits
- Introduction to limits at infinity
- Limits at infinity of quotients
- Limits at infinity of quotients (Part 1)
- Limits at infinity of quotients (Part 2)
- Limits at infinity of quotients with square roots
- Limits at infinity of quotients with square roots (even power)
- Limits at infinity of quotients with square roots (odd power)
- Limits at infinity: graphical
- Limits by factoring
- Limits by factoring
- Limits intro
- Limits intro
- Limits of combined functions: piecewise functions
- Limits of piecewise functions
- Limits of piecewise functions
- Limits of piecewise functions: absolute value
- Limits of trigonometric functions
- Limits of trigonometric functions
- Limits using trig identities
- Trig limit using double angle identity
- Trig limit using Pythagorean identity
Determine the limit of a function numerically and verify with graphing technology.
Use proper limit notation, including when describing the end behavior of a function.
- Functions with same limit at infinity
- Introduction to limits at infinity
- Limits at infinity of quotients
- Limits at infinity of quotients (Part 1)
- Limits at infinity of quotients (Part 2)
- Limits at infinity of quotients with square roots
- Limits at infinity of quotients with square roots (even power)
- Limits at infinity of quotients with square roots (odd power)
- Limits at infinity: graphical
determine the limit of a function numerically by direct substitution;
- Conclusions from direct substitution (finding limits)
- Direct substitution with limits that don't exist
- Limits and continuity: FAQ
- Limits by direct substitution
- Limits by direct substitution
- Limits intro
- Limits intro
- Limits of trigonometric functions
- Limits of trigonometric functions
- Limits using trig identities
- Strategy in finding limits
- Strategy in finding limits
- Strategy in finding limits
- Undefined limits by direct substitution
determine the limit of a function using algebraic manipulation;
- Infinite limits: algebraic
- Limit of (1-cos(x))/x as x approaches 0
- Limit of sin(x)/x as x approaches 0
- Limits by factoring
- Limits by factoring
- Limits by rationalizing
- Limits intro
- Limits intro
- Limits of combined functions: piecewise functions
- Limits of piecewise functions
- Limits of piecewise functions
- Limits of piecewise functions: absolute value
- Limits of trigonometric functions
- Limits of trigonometric functions
- Limits using conjugates
- Limits using trig identities
- Next steps after indeterminate form (finding limits)
- Strategy in finding limits
- Trig limit using double angle identity
- Trig limit using Pythagorean identity
estimate the limit of a function using a table; and
- Analyzing unbounded limits: mixed function
- Analyzing unbounded limits: rational function
- Approximating limits using tables
- Creating tables for approximating limits
- Estimating limits from tables
- Estimating limits from tables
- Limits and continuity: FAQ
- Limits intro
- One-sided limits from tables
- One-sided limits from tables
- Using tables to approximate limit values
determine the limit of a function from a given graph.
- Connecting limits and graphical behavior
- Connecting limits and graphical behavior
- Estimating limit values from graphs
- Estimating limit values from graphs
- Estimating limit values from graphs
- Infinite limits and asymptotes
- Infinite limits: graphical
- Introduction to infinite limits
- Introduction to limits at infinity
- Limits and continuity: FAQ
- Limits at infinity: graphical
- Limits intro
- Limits intro
- Limits of combined functions
- Limits of combined functions: piecewise functions
- Limits of combined functions: products and quotients
- Limits of combined functions: sums and differences
- Limits of composite functions
- Limits of composite functions: external limit doesn't exist
- Limits of composite functions: internal limit doesn't exist
- One-sided limits from graphs
- One-sided limits from graphs
- One-sided limits from graphs: asymptote
- Theorem for limits of composite functions
- Theorem for limits of composite functions: when conditions aren't met
- Unbounded limits
As the variable approaches positive or negative infinity, analyze the limit of a function to describe the end behavior.
- Functions with same limit at infinity
- Introduction to limits at infinity
- Limits at infinity of quotients
- Limits at infinity of quotients (Part 1)
- Limits at infinity of quotients (Part 2)
- Limits at infinity of quotients with square roots
- Limits at infinity of quotients with square roots (even power)
- Limits at infinity of quotients with square roots (odd power)
- Limits at infinity: graphical
Describe continuity of a function.
- Continuity and common functions
- Continuity at a point
- Continuity at a point (algebraic)
- Continuity at a point (graphical)
- Continuity over an interval
- Continuity over an interval
- Functions continuous at specific x-values
- Functions continuous on all real numbers
- Limits and continuity: FAQ
- Worked example: Continuity at a point (graphical)
- Worked example: point where a function is continuous
- Worked example: point where a function isn't continuous
Use mathematical notation to communicate and describe the continuity of functions including polynomial, rational, piecewise, absolute value, radical, and step function, using graphical and algebraic methods.
- Continuity and common functions
- Continuity at a point
- Continuity at a point (algebraic)
- Continuity at a point (graphical)
- Continuity over an interval
- Functions continuous at specific x-values
- Functions continuous on all real numbers
- Worked example: Continuity at a point (graphical)
- Worked example: point where a function is continuous
- Worked example: point where a function isn't continuous
Prove continuity at a point, using the definition.
Classify types of discontinuity based on which condition of continuity is violated.