Questions and Answers
What is a hyperbola?
An open curve formed by a plane that cuts the base of a right circular cone.
What is the hyperbola equation centered at (h,k)?
Center = (h,k), a = distance from the center to build the box for asymptotes, b = distance from the center to build the box for asymptotes.
What is an ellipse?
A regular oval shape where the sum of distances from two points is constant.
What is the ellipse equation centered at (h,k)?
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What is a circle?
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What is the circle equation centered at (h,k)?
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What is a parabola?
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What is the parabola equation in vertex form?
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What is a quadratic equation?
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What is the quadratic formula used for?
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What are conic sections?
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What does the slope of a line represent?
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What is the slope-intercept form of a line?
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The slope of the line y=1 is vertical.
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The slope of the line x=-4 is horizontal.
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What characteristic do parallel lines share regarding slope?
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What is the relationship between the slopes of perpendicular lines?
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A line with a negative slope rises from left to right.
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A line with a positive slope falls from left to right.
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What is an imaginary number?
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What is the value of i^2?
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What is the value of i^3?
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What is the value of i^4?
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What is a discriminant?
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What is a complex number?
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What is the complex plane?
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What does the Pythagorean Theorem state?
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What is completing the square?
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What are conjugates?
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A false equation has values that equal each other on both sides.
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An identity equation is true for all real numbers.
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What is an asymptote?
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What is the distance equation?
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The expression am + an + bm + bn = ___
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(x^2)^4 = ___
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(x^4)/(x^2) = ___
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(x^4)(x^2) = ___
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X^(2/4) = ___
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(x + a)^2 = ___
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(x - a)^2 = ___
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(x + a)(x - a) = ___
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16x^4 - 25y^6 = ___
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137^0 = ___
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The square root of 0 is ___
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What is a system of equations?
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Study Notes
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.
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Test your understanding of key concepts in Algebra 2 with these flashcards. This quiz covers important terms like hyperbola and hyperbola equations, helping reinforce your knowledge of conic sections. Perfect for students using Teaching Textbooks curriculum.