Hlavní obsah
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.
(Content unavailable)
domain and range;
(Content unavailable)
symmetry (including even and odd functions);
relative and/or absolute maximum and minimum points.
(Content unavailable)
asymptotes (horizontal, vertical, and oblique/slant;
- Discontinuities 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 function points of discontinuity
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 function points of discontinuity
- 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;
(Content unavailable)
end behavior; and
(Content unavailable)
Sketch the graph of a polynomial, rational, piecewise-defined, absolute value, radical, and step function.
- Absolute value graphs review
- Graph absolute value functions
- Graphing absolute value functions
- Graphing rational functions according to asymptotes
- Graphing square and cube root functions
- 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
- Radical functions & their graphs
- Radical functions & their graphs
- Shifting absolute value graphs
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 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 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 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
- 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: 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
- 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.