Teaching Textbooks Algebra 2 Flashcards

Conic Sections

• A hyperbola is an open curve formed by the intersection of a plane and a right circular cone.
• The hyperbola equation includes a center at (h,k) with parameters 'a' and 'b' indicating distances to build asymptotes, resembling the ellipse equation with a negative difference.
• An ellipse is a regular oval shape defined by a constant sum of distances from two focal points or by cutting a cone with an oblique plane.
• The ellipse equation features a center at (h,k) where 'a' denotes the distance along the x-axis and 'b' along the y-axis.
• A circle is defined as a round plane figure with all points equidistant from the center.

Equations of Curves

• The circle equation is centered at (h,k) with radius 'r'.
• A parabola graphically represents a quadratic function.
• The parabola equation includes a vertex at (h,k) and parameter 'a', which indicates the direction (up or down) and the width of the parabola.
• Quadratic equations take the form ax² + bx + c = 0, where 'a' is non-zero.
• The quadratic formula is employed to solve difficult quadratic equations that do not factor easily.

Line Properties

• Conic sections include parabolas, circles, ellipses, and hyperbolas formed by intersections of a plane with a double cone.
• The slope of a line is determined by the ratio of rise/run or change in y/change in x.
• Slope-intercept form is expressed as y = mx + b, where 'm' is slope and 'b' is the y-intercept.
• A line with a slope of 1 (y=1) is horizontal with zero slope; a line x=-4 is vertical with an undefined slope.
• Parallel lines have identical slopes, while perpendicular lines have slopes that are negative reciprocals of one another.

Slope Characteristics

• A negative slope indicates a line falling from left to right; a positive slope indicates a line rising in the same direction.
• Imaginary numbers arise as square roots of negative numbers, with 'i' defined as √-1.
• Powers of 'i' include i² = -1, i³ = -i, and i⁴ = 1.

Advanced Concepts

• The discriminant of a polynomial provides insight into its roots.
• Complex numbers of the form a + bi consist of real numbers 'a' and 'b', with 'i' representing the square root of -1.
• The complex plane represents a coordinate system with real and imaginary axes.

Algebraic Principles

• The Pythagorean theorem is represented as a² + b² = c².
• Completing the square transform a quadratic expression into a perfect square trinomial.
• Conjugates align as pairs of expressions a+√b and a-√b, where 'a' and 'b' are rational.
• A false equation holds unequal values on either side of the equals sign, while an identity equation is always true.

Asymptotes and Distance

• Asymptotes approach curves but never intersect them within finite distances.
• The distance equation utilizes the Pythagorean theorem variant to calculate the distance between two points on a graph.

Algebraic Identities

• The equation am + an + bm + bn is factored as (a+b)(m+n).
• Power rules indicate (x²)⁴ = x⁸ and (x⁴)/(x²) = x².
• Multiplications yield (x⁴)(x²) = x⁶, and fractional powers express x^(2/4) as the 4th root of x².
• Squared binomials expand to (x+a)² = x²+2ax+a² and (x-a)² = x²-2ax+a².
• The difference of squares identity states (x+a)(x-a) = x²-a².
• For 16x⁴-25y⁶, factoring yields (4x²+5y³)(4x²-5y³).
• Any non-zero number raised to the power of zero equals 1, while the square root of 0 equals 0.