Systems Matrices and Conic Sections flashcards

0/36

36 cards remaining

Take the chapter practice test

All 36 cards in this deckShow

What is common solution?Systems of Equations in Two and Three Variables: A simultaneous assignment of unknowns.
Does satisfying one equation suffice?Systems of Equations in Two and Three Variables: No, check every equation in the system.
What is elimination?Systems of Equations in Two and Three Variables: A variable-removal method for systems.
Why scale every term of a row?Systems of Equations in Two and Three Variables: Scaling only part of an equation changes its solutions.
What is consistency and meaning?Systems of Equations in Two and Three Variables: Classification of simultaneous constraints.
Why check contextual restrictions?Systems of Equations in Two and Three Variables: A mathematical answer may be impossible in the real problem.
What is augmented matrix?Solving Systems with Matrices: A coefficient table with a separated constant column.
Why keep variable order fixed?Solving Systems with Matrices: A swapped column would represent a different system.
What is elementary row operations?Solving Systems with Matrices: Reversible transformations of a linear system.
Why prohibit scaling a row by zero?Solving Systems with Matrices: It destroys information and cannot be reversed.
What is back-substitution?Solving Systems with Matrices: Recovering unknowns from an equivalent row-reduced system.
Why check the original equations?Solving Systems with Matrices: Entry or arithmetic errors can occur during row operations.
What is center-radius form?Circles: An equation for points a fixed distance from a center.
Why take a positive square root for r?Circles: Radius is a nonnegative distance.
What is completing the square?Circles: An algebraic rewrite from expanded circle equation.
Why balance added square terms?Circles: Adding to only one side changes the circle.
What is coordinate distance?Circles: The straight-line distance between two plane points.
Can radius be negative?Circles: No, a distance cannot be negative.
What is focus and directrix?Parabolas as Loci: A conic locus defined by equal distances.
Is the vertex halfway between focus and directrix?Parabolas as Loci: Yes, along the symmetry axis.
What is vertical standard form?Parabolas as Loci: A focus-directrix equation for a vertical parabola.
What does negative p mean?Parabolas as Loci: The parabola opens downward.
What is horizontal standard form?Parabolas as Loci: A focus-directrix equation for a horizontal parabola.
How does p affect its direction?Parabolas as Loci: Positive p opens right; negative p opens left.
What is focal definition?Ellipses: A conic defined by a fixed sum of focal distances.
Is every circle also an ellipse?Ellipses: Yes, with its two foci at the center.
What is standard axes?Ellipses: Major and minor semiaxes read from standard form.
Which denominator marks the major axis?Ellipses: The larger positive squared semiaxis.
What is focal distance?Ellipses: The distance from center to each ellipse focus.
Could focal distance exceed the major semiaxis?Ellipses: No, because f²=A²−B²<A².
What is hyperbola definition?Hyperbolas and Conic Comparisons: A conic with two separate branches.
How does its focal condition differ from an ellipse?Hyperbolas and Conic Comparisons: It uses an absolute difference rather than a sum.
What is asymptotes and orientation?Hyperbolas and Conic Comparisons: Lines describing distant branch directions.
What changes if y² has the positive term?Hyperbolas and Conic Comparisons: The branches open up and down.
What is conic identification?Hyperbolas and Conic Comparisons: Classifying basic axis-aligned conic equations.
Can coefficients alone determine every graph without checking constants?Hyperbolas and Conic Comparisons: No, degeneracies and empty real sets are possible.