Quadratic Equations and Functions flashcards

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Do not divide away a rootSolving Quadratics by Factoring: Factoring keeps zero visible; dividing by x can erase it.
When should I use zero-product solutions?Solving Quadratics by Factoring: Solve a quadratic already written as a product.
What method supports do not divide away a root?Solving Quadratics by Factoring: x² − kx = x(x − k)
Factor before solvingSolving Quadratics by Factoring: First make one side zero, then factor and solve.
Zero-product solutionsSolving Quadratics by Factoring: A zero product means one of its factors is zero.
What must I check in solving quadratics by factoring?Solving Quadratics by Factoring: The zero-product property applies to a product equal to zero; list all real solutions
What method supports factor before solving?Solving Quadratics by Factoring: x² − (a+b)x + ab = 0 ⇒ x = a or b
What method supports zero-product solutions?Solving Quadratics by Factoring: AB = 0 ⇒ A = 0 or B = 0
When should I use do not divide away a root?Solving Quadratics by Factoring: Preserve a possible zero solution during factoring.
When should I use factor before solving?Solving Quadratics by Factoring: Move all terms to one side and factor.
Complete the square to solveSquare Roots and Completing the Square: Create a square, solve it, and undo its shift.
When should I use square-root equations?Square Roots and Completing the Square: Solve an equation with a squared binomial.
What method supports complete the square to solve?Square Roots and Completing the Square: x² + 2hx = c ⇒ (x + h)² = c + h²
Complete a monic squareSquare Roots and Completing the Square: Halve the x-coefficient and square it.
Square-root equationsSquare Roots and Completing the Square: A positive square has two possible signed roots.
What must I check in square roots and completing the square?Square Roots and Completing the Square: Include both square-root signs when solving a square equal to a positive number; keep real-domain conditions
What method supports complete a monic square?Square Roots and Completing the Square: x² + bx + (b/2)² = (x + b/2)²
What method supports square-root equations?Square Roots and Completing the Square: (x − h)² = k ⇒ x = h ± √k for k ≥ 0
When should I use complete the square to solve?Square Roots and Completing the Square: Shift the constant and isolate a square in a nonfactored equation.
When should I use complete a monic square?Square Roots and Completing the Square: Choose the constant that creates a perfect square.
Parameter at the repeated-root boundaryQuadratic Formula and Discriminant: At the boundary between two roots and no roots, the discriminant is zero.
When should I use quadratic formula?Quadratic Formula and Discriminant: Use standard-form coefficients in the quadratic formula.
What method supports parameter at the repeated-root boundary?Quadratic Formula and Discriminant: b² − 4ac = 0 for a repeated real root
Discriminant classificationQuadratic Formula and Discriminant: The sign under the square root controls the real solutions.
Quadratic formulaQuadratic Formula and Discriminant: Keep the signs and the whole denominator.
What must I check in quadratic formula and discriminant?Quadratic Formula and Discriminant: Use a ≠ 0; the discriminant classifies real roots, and a repeated root counts once
What method supports discriminant classification?Quadratic Formula and Discriminant: D > 0: two; D = 0: one; D < 0: none
What method supports quadratic formula?Quadratic Formula and Discriminant: x = [−b ± √(b² − 4ac)]/(2a)
When should I use parameter at the repeated-root boundary?Quadratic Formula and Discriminant: Set the discriminant equal to zero to find a tangency condition.
When should I use discriminant classification?Quadratic Formula and Discriminant: Use the discriminant to count real roots.
Convert standard to vertex formStandard, Factored, and Vertex Forms: Make a square without changing the function value.
When should I use read vertex form?Standard, Factored, and Vertex Forms: Identify vertex coordinates and evaluate a vertex-form function.
What method supports convert standard to vertex form?Standard, Factored, and Vertex Forms: x² − 2hx + c = (x − h)² + c − h²
Convert factored formStandard, Factored, and Vertex Forms: Zeros are visible in factors; distribute to recover coefficients.
Read vertex formStandard, Factored, and Vertex Forms: Inside locates left-right; outside locates up-down.
What must I check in standard, factored, and vertex forms?Standard, Factored, and Vertex Forms: Equivalent forms describe the same function and preserve all coefficients
What method supports convert factored form?Standard, Factored, and Vertex Forms: a(x − r)(x − s) = ax² − a(r+s)x + ars
What method supports read vertex form?Standard, Factored, and Vertex Forms: f(x) = a(x − h)² + k has vertex (h,k)
When should I use convert standard to vertex form?Standard, Factored, and Vertex Forms: Complete a square to determine a vertex value.
When should I use convert factored form?Standard, Factored, and Vertex Forms: Expand factored form to read a standard-form coefficient.
Recover a graph parameterRoots, Intercepts, Symmetry, and Graphs: The roots locate the crossings; another point sets the vertical scale.
When should I use roots and intercepts?Roots, Intercepts, Symmetry, and Graphs: Read zeros from a factored quadratic.
What method supports recover a graph parameter?Roots, Intercepts, Symmetry, and Graphs: y₀ = a(x₀ − r)(x₀ − s)
Axis of symmetryRoots, Intercepts, Symmetry, and Graphs: The symmetry line is halfway between the roots.
Roots and interceptsRoots, Intercepts, Symmetry, and Graphs: Set y to zero for the horizontal-axis crossings.
What must I check in roots, intercepts, symmetry, and graphs?Roots, Intercepts, Symmetry, and Graphs: Real roots correspond to x-intercepts; the axis of symmetry is vertical
What method supports axis of symmetry?Roots, Intercepts, Symmetry, and Graphs: h = (r + s)/2 = −b/(2a)
What method supports roots and intercepts?Roots, Intercepts, Symmetry, and Graphs: a(x − r)(x − s) = 0 ⇒ x = r or s
When should I use recover a graph parameter?Roots, Intercepts, Symmetry, and Graphs: Use one point to determine a quadratic multiplier.
When should I use axis of symmetry?Roots, Intercepts, Symmetry, and Graphs: Average the roots to find the axis of a parabola.
Optimization from standard formMaximum and Minimum Values: Find the vertex input, then calculate the output there.
When should I use minimum in vertex form?Maximum and Minimum Values: Find the least value of an upward-opening quadratic.
What method supports optimization from standard form?Maximum and Minimum Values: h = −b/(2a), extreme output = f(h)
Maximum in vertex formMaximum and Minimum Values: A negative square term can only subtract from the top value.
Minimum in vertex formMaximum and Minimum Values: The squared part cannot go below zero.
What must I check in maximum and minimum values?Maximum and Minimum Values: A vertex minimum requires a > 0; a maximum requires a < 0; restrict a contextual domain when stated
What method supports maximum in vertex form?Maximum and Minimum Values: a < 0 ⇒ maximum of a(x − h)² + k is k
What method supports minimum in vertex form?Maximum and Minimum Values: a > 0 ⇒ minimum of a(x − h)² + k is k
When should I use optimization from standard form?Maximum and Minimum Values: Complete a square or use the vertex formula to optimize.
When should I use maximum in vertex form?Maximum and Minimum Values: Find the greatest value of a downward-opening quadratic.
Revenue modelingQuadratic Modeling and Word Problems: Multiply price by sales, then optimize the product.
When should I use area models?Quadratic Modeling and Word Problems: Build a quadratic from two related dimensions.
What method supports revenue modeling?Quadratic Modeling and Word Problems: R(p) = p(A − Bp)
Height over timeQuadratic Modeling and Word Problems: The top of the parabola gives the peak height.
Area modelsQuadratic Modeling and Word Problems: Multiply the two side expressions, then use the area.
What must I check in quadratic modeling and word problems?Quadratic Modeling and Word Problems: Reject roots outside the physical domain; state units for lengths, times, areas, and output values
What method supports height over time?Quadratic Modeling and Word Problems: H(t) = −at² + bt + c ⇒ t_peak = b/(2a), a > 0
What method supports area models?Quadratic Modeling and Word Problems: A = width × length
When should I use revenue modeling?Quadratic Modeling and Word Problems: Optimize a product of price and demand.
When should I use height over time?Quadratic Modeling and Word Problems: Interpret a quadratic vertex in a motion model.
Parameter interval for real rootsUnknown Coefficients and Conditions on Solutions: Translate the root-count condition into a condition on the discriminant.
When should I use specified root constraint?Unknown Coefficients and Conditions on Solutions: Find a coefficient from a stated root.
What method supports parameter interval for real roots?Unknown Coefficients and Conditions on Solutions: b² − 4ac ≥ 0 for real roots
Root sums and productsUnknown Coefficients and Conditions on Solutions: Roots encode the middle and constant coefficients.
Specified root constraintUnknown Coefficients and Conditions on Solutions: Put the known root into the equation and make the result zero.
What must I check in unknown coefficients and conditions on solutions?Unknown Coefficients and Conditions on Solutions: Check a ≠ 0 before applying quadratic conditions; distinguish a specified root from a repeated root
What method supports root sums and products?Unknown Coefficients and Conditions on Solutions: r + s = −b/a; rs = c/a
What method supports specified root constraint?Unknown Coefficients and Conditions on Solutions: r root of ax² + bx + c ⇒ ar² + br + c = 0
When should I use parameter interval for real roots?Unknown Coefficients and Conditions on Solutions: Solve a discriminant inequality for a parameter.
When should I use root sums and products?Unknown Coefficients and Conditions on Solutions: Relate unknown coefficients to the roots.