Other Nonlinear Equations and Functions flashcards

0/80

80 cards remaining

Take the chapter practice test

All 80 cards in this deckShow

Nested linear absolute equationsAbsolute-Value Equations and Graphs: Unwrap the outside operations before creating the two cases.
When should I use distance equations?Absolute-Value Equations and Graphs: Solve an absolute-value equation as two signed cases.
What method supports nested linear absolute equations?Absolute-Value Equations and Graphs: a|bx + c| + d = e ⇒ |bx+c| = (e−d)/a, a ≠ 0
Absolute-value graph vertexAbsolute-Value Equations and Graphs: The corner occurs where the inside of the absolute value is zero.
Distance equationsAbsolute-Value Equations and Graphs: Distance can come from either side of zero.
What must I check in absolute-value equations and graphs?Absolute-Value Equations and Graphs: Absolute value is nonnegative; a positive right side usually gives two branches, zero one, and a negative none
What method supports absolute-value graph vertex?Absolute-Value Equations and Graphs: y = a|x − h| + k has vertex (h,k)
What method supports distance equations?Absolute-Value Equations and Graphs: |x − h| = k ⇒ x = h ± k, k ≥ 0
When should I use nested linear absolute equations?Absolute-Value Equations and Graphs: Isolate absolute value before splitting into cases.
When should I use absolute-value graph vertex?Absolute-Value Equations and Graphs: Read the vertex of a shifted absolute-value graph.
Radical in a shifted expressionRadical Equations and Extraneous Solutions: Call the root t, but remember t cannot be negative.
When should I use isolated radical?Radical Equations and Extraneous Solutions: Square a nonnegative radical equation and verify.
What method supports radical in a shifted expression?Radical Equations and Extraneous Solutions: t = √x ⇒ t ≥ 0 and x = t²
Extraneous root testRadical Equations and Extraneous Solutions: A squared equation can accept a value the original rejects.
Isolated radicalRadical Equations and Extraneous Solutions: Check the signs, square, solve, and check again.
What must I check in radical equations and extraneous solutions?Radical Equations and Extraneous Solutions: Squaring can create extraneous candidates; all answers must satisfy the original root equation
What method supports extraneous root test?Radical Equations and Extraneous Solutions: √u = v requires u ≥ 0 and v ≥ 0
What method supports isolated radical?Radical Equations and Extraneous Solutions: √(x+a)=b ⇒ x=b²−a when b≥0
When should I use radical in a shifted expression?Radical Equations and Extraneous Solutions: Use a substitution to preserve the square-root sign.
When should I use extraneous root test?Radical Equations and Extraneous Solutions: Discard a squared-equation candidate that fails the original.
Canceled expressions still have restrictionsRational Equations and Excluded Values: A hole stays forbidden after cancellation.
When should I use single rational equation?Rational Equations and Excluded Values: Clear a denominator and check its excluded value.
What method supports canceled expressions still have restrictions?Rational Equations and Excluded Values: (x²−a²)/(x−a)=x+a only for x≠a
Equations with common denominatorsRational Equations and Excluded Values: Clear every fraction at once, then check the original domain.
Single rational equationRational Equations and Excluded Values: Write the forbidden value before clearing the fraction.
What must I check in rational equations and excluded values?Rational Equations and Excluded Values: Every original denominator must be nonzero before or after clearing fractions
What method supports equations with common denominators?Rational Equations and Excluded Values: a/x + b/x = c ⇒ x=(a+b)/c, x≠0, c≠0
What method supports single rational equation?Rational Equations and Excluded Values: a/(x−h)=b ⇒ x=h+a/b for b≠0 and x≠h
When should I use canceled expressions still have restrictions?Rational Equations and Excluded Values: Identify a rational equation with an excluded algebraic candidate.
When should I use equations with common denominators?Rational Equations and Excluded Values: Use a common multiplier without losing restrictions.
Remainder and evaluationPolynomial Equations, Zeros, and Graphs: A factor condition is an evaluation equal to zero.
When should I use polynomial zeros?Polynomial Equations, Zeros, and Graphs: Read zeros from multiple factors.
What method supports remainder and evaluation?Polynomial Equations, Zeros, and Graphs: x−r divides P(x) ⇔ P(r)=0
Multiplicity and crossingPolynomial Equations, Zeros, and Graphs: Odd multiplicity crosses; even multiplicity touches.
Polynomial zerosPolynomial Equations, Zeros, and Graphs: Each factor supplies its own zero.
What must I check in polynomial equations, zeros, and graphs?Polynomial Equations, Zeros, and Graphs: Count distinct roots separately from multiplicities; zeros of denominators are not polynomial roots
What method supports multiplicity and crossing?Polynomial Equations, Zeros, and Graphs: even multiplicity: no sign change; odd multiplicity: sign change
What method supports polynomial zeros?Polynomial Equations, Zeros, and Graphs: a∏(x−rᵢ)=0 ⇒ at least one x=rᵢ
When should I use remainder and evaluation?Polynomial Equations, Zeros, and Graphs: Use a polynomial value to constrain an unknown constant.
When should I use multiplicity and crossing?Polynomial Equations, Zeros, and Graphs: Determine whether a graph crosses or touches at a root.
Combined domain restrictionsFunction Notation, Evaluation, and Domain: Every part of the formula must be defined at the same input.
When should I use function inputs?Function Notation, Evaluation, and Domain: Evaluate a function at a number or expression.
What method supports combined domain restrictions?Function Notation, Evaluation, and Domain: √(x−a)/(x−b): x≥a and x≠b
Domain of a radicalFunction Notation, Evaluation, and Domain: Keep the inside of a square root at least zero.
Function inputsFunction Notation, Evaluation, and Domain: Substitute the input into every occurrence of the variable.
What must I check in function notation, evaluation, and domain?Function Notation, Evaluation, and Domain: Inputs must satisfy every original denominator and even-root restriction
What method supports domain of a radical?Function Notation, Evaluation, and Domain: √(ax+b) real ⇒ ax+b ≥ 0
What method supports function inputs?Function Notation, Evaluation, and Domain: If f(x)=x²+c, then f(a)=a²+c
When should I use combined domain restrictions?Function Notation, Evaluation, and Domain: Intersect radical and denominator conditions.
When should I use domain of a radical?Function Notation, Evaluation, and Domain: Determine the allowed inputs for an even-root function.
Equivalent nonlinear representationsFunction Transformations and Equivalent Representations: Expand or complete the square to show the formulas match.
When should I use vertical transformations?Function Transformations and Equivalent Representations: Evaluate a vertically stretched and shifted function.
What method supports equivalent nonlinear representations?Function Transformations and Equivalent Representations: a(x−h)²+k = ax²−2ahx+ah²+k
Horizontal shifts and scalingFunction Transformations and Equivalent Representations: Solve for the new input that reproduces the old argument.
Vertical transformationsFunction Transformations and Equivalent Representations: Change the outputs after evaluating the original function.
What must I check in function transformations and equivalent representations?Function Transformations and Equivalent Representations: Horizontal changes act inside the input; vertical changes act on the output
What method supports horizontal shifts and scaling?Function Transformations and Equivalent Representations: If (u,v) is on f, then (u+h,v) is on f(x−h) and (u/c,v) is on f(cx), c≠0
What method supports vertical transformations?Function Transformations and Equivalent Representations: g(x)=af(x)+k
When should I use equivalent nonlinear representations?Function Transformations and Equivalent Representations: Compare expanded and transformed expressions.
When should I use horizontal shifts and scaling?Function Transformations and Equivalent Representations: Track a known point through an input shift or scale.
Systems with a squared relationSystems Involving Linear and Nonlinear Equations: Solve one variable, then rebuild each complete pair.
When should I use line and parabola intersections?Systems Involving Linear and Nonlinear Equations: Substitute a line into a quadratic relation.
What method supports systems with a squared relation?Systems Involving Linear and Nonlinear Equations: x=y+c and x=y² ⇒ y²−y−c=0
A line through a shifted parabolaSystems Involving Linear and Nonlinear Equations: A horizontal line fixes the output and leaves you to solve for inputs.
Line and parabola intersectionsSystems Involving Linear and Nonlinear Equations: Set the two outputs equal to locate their shared points.
What must I check in systems involving linear and nonlinear equations?Systems Involving Linear and Nonlinear Equations: Check ordered pairs in both originals; a two-root equation can give two intersection points
What method supports a line through a shifted parabola?Systems Involving Linear and Nonlinear Equations: (x−h)² = r² ⇒ x=h±r
What method supports line and parabola intersections?Systems Involving Linear and Nonlinear Equations: y=f(x), y=g(x) ⇒ f(x)=g(x)
When should I use systems with a squared relation?Systems Involving Linear and Nonlinear Equations: Use substitution while preserving multiple sign possibilities.
When should I use a line through a shifted parabola?Systems Involving Linear and Nonlinear Equations: Solve a nonlinear system after substitution.
Feasible nonlinear branchesNonlinear Parameters and Number of Intersections: Find all candidates and then filter by the permitted inputs.
When should I use horizontal level intersections?Nonlinear Parameters and Number of Intersections: Compare a horizontal level with a parabola extremum.
What method supports feasible nonlinear branches?Nonlinear Parameters and Number of Intersections: feasible solutions = algebraic solutions ∩ allowed domain
Tangency parameterNonlinear Parameters and Number of Intersections: One distinct crossing location occurs when the quadratic discriminant is zero.
Horizontal level intersectionsNonlinear Parameters and Number of Intersections: Above the bottom gives two crossings; at the bottom gives one.
What must I check in nonlinear parameters and number of intersections?Nonlinear Parameters and Number of Intersections: Intersection counts must respect domains; tangency corresponds to one distinct solution rather than two repeated labels
What method supports tangency parameter?Nonlinear Parameters and Number of Intersections: D=0 for one real quadratic intersection
What method supports horizontal level intersections?Nonlinear Parameters and Number of Intersections: (x−h)²=c−k
When should I use feasible nonlinear branches?Nonlinear Parameters and Number of Intersections: Count only intersections on an allowed interval.
When should I use tangency parameter?Nonlinear Parameters and Number of Intersections: Use a discriminant to identify a line touching a parabola.