Teaching Textbooks Algebra 2 Flashcards
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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)?

<p>Center = (h,k), a = x-axis distance from the center, b = y-axis distance from the center.</p> Signup and view all the answers

What is a circle?

<p>A round plane figure whose boundary consists of points equidistant from the center.</p> Signup and view all the answers

What is the circle equation centered at (h,k)?

<p>Center = (h,k), radius = r.</p> Signup and view all the answers

What is a parabola?

<p>The shape of the graph of a quadratic function.</p> Signup and view all the answers

What is the parabola equation in vertex form?

<p>Vertex = (h,k), a = width orientation.</p> Signup and view all the answers

What is a quadratic equation?

<p>An equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$.</p> Signup and view all the answers

What is the quadratic formula used for?

<p>Used to solve quadratic equations that don't factor easily.</p> Signup and view all the answers

What are conic sections?

<p>Four shapes: parabolas, circles, ellipses, and hyperbolas formed by the intersection of a plane and a double cone.</p> Signup and view all the answers

What does the slope of a line represent?

<p>Rise/run or change in y/change in x.</p> Signup and view all the answers

What is the slope-intercept form of a line?

<p>y = mx + b, where m is the slope and b is the y-intercept.</p> Signup and view all the answers

The slope of the line y=1 is vertical.

<p>False</p> Signup and view all the answers

The slope of the line x=-4 is horizontal.

<p>False</p> Signup and view all the answers

What characteristic do parallel lines share regarding slope?

<p>Parallel lines have the same slope.</p> Signup and view all the answers

What is the relationship between the slopes of perpendicular lines?

<p>They are negative reciprocals of each other.</p> Signup and view all the answers

A line with a negative slope rises from left to right.

<p>False</p> Signup and view all the answers

A line with a positive slope falls from left to right.

<p>False</p> Signup and view all the answers

What is an imaginary number?

<p>The square root of a negative number.</p> Signup and view all the answers

What is the value of i^2?

<p>-1.</p> Signup and view all the answers

What is the value of i^3?

<p>-i.</p> Signup and view all the answers

What is the value of i^4?

<ol> <li></li> </ol> Signup and view all the answers

What is a discriminant?

<p>A function of the coefficients of a polynomial equation that provides information about the roots.</p> Signup and view all the answers

What is a complex number?

<p>A number of the form a + bi where a and b are real numbers.</p> Signup and view all the answers

What is the complex plane?

<p>A set of coordinate axes with the horizontal axis as the real axis and the vertical as the imaginary axis.</p> Signup and view all the answers

What does the Pythagorean Theorem state?

<p>a² + b² = c².</p> Signup and view all the answers

What is completing the square?

<p>A process to make a quadratic expression into a perfect square trinomial.</p> Signup and view all the answers

What are conjugates?

<p>The expressions a + √b and a - √b where a and b are rational numbers.</p> Signup and view all the answers

A false equation has values that equal each other on both sides.

<p>False</p> Signup and view all the answers

An identity equation is true for all real numbers.

<p>True</p> Signup and view all the answers

What is an asymptote?

<p>A line that continually approaches a given curve but does not meet it at any finite distance.</p> Signup and view all the answers

What is the distance equation?

<p>A variation on the Pythagorean theorem to find the distance between two points on a graph.</p> Signup and view all the answers

The expression am + an + bm + bn = ___

<p>(a + b)(m + n).</p> Signup and view all the answers

(x^2)^4 = ___

<p>x^8.</p> Signup and view all the answers

(x^4)/(x^2) = ___

<p>x^2.</p> Signup and view all the answers

(x^4)(x^2) = ___

<p>x^6.</p> Signup and view all the answers

X^(2/4) = ___

<p>4th root of x^2.</p> Signup and view all the answers

(x + a)^2 = ___

<p>x^2 + 2ax + a^2.</p> Signup and view all the answers

(x - a)^2 = ___

<p>x^2 - 2ax + a^2.</p> Signup and view all the answers

(x + a)(x - a) = ___

<p>x^2 - a^2.</p> Signup and view all the answers

16x^4 - 25y^6 = ___

<p>(4x^2 + 5y^3)(4x^2 - 5y^3).</p> Signup and view all the answers

137^0 = ___

<ol> <li></li> </ol> Signup and view all the answers

The square root of 0 is ___

<ol start="0"> <li></li> </ol> Signup and view all the answers

What is a system of equations?

<p>A set of equations with the same variables.</p> Signup and view all the answers

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.

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