PRE-CAL

• Conic Section


  Particular class of curves which often times appear in nature and which have applications in other fields. The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone
• Types of Conic Sections

  • Circle
  • Ellipse
  • Parabola
  • Hyperbola
• Degenerate Conic Section


  A degenerate conic is generated when a plane intersects the vertex of the cone. There are three types of degenerate conics: the degenerate form of a circle or an ellipse is a singular point, the degenerate form of a parabola is a line, the degenerate form of a hyperbola is two intersecting lines
• Equation of Circle


  The standard equation of a circle with center at (0,0) is x^2 + y^2 = r^2, and the standard equation of a circle with center at (h,k) is (x-h)^2 + (y-k)^2 = r^2
• Parabola


  A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola
• Parts of a Parabola

  • Vertex
  • Focus
  • Directrix
  • Axis of Symmetry
  • Latus Rectum
• Standard Equations of Parabola

  • Parabola opening to the Right: y = ax^2
  • Parabola opening to the Left: x = -4ay
  • Parabola opening Upwards: x = ay^2
  • Parabola opening Downwards: y = -ax^2
• Ellipse


  The set of all points in a plane, the sum of whose distances from two fixed points, called foci, is a constant
• Parts of an Ellipse

  • Major Axis
  • Minor Axis
  • Center
  • Vertices
  • Co-Vertices
  • Foci
  • Latera Recta
• Standard Equation of Ellipse with Center at (0,0)

  x^2/a^2 + y^2/b^2 = 1
• Standard Equation of Ellipse with Center at (h,k)

  (x-h)^2/a^2 + (y-k)^2/b^2 = 1
• Hyperbola


  The standard equation of a hyperbola with center at (0,0) and transverse axis along the x-axis is (x^2/a^2) - (y^2/b^2) = 1, and the standard equation with center at (h,k) and transverse axis along the x-axis is ((x-h)^2/a^2) - ((y-k)^2/b^2) = 1
• General Equation of Conic Section


  Ax^2 + By^2 + Cx + Dy + E = 0
• System of Nonlinear Equations

  A system of two or more equations in two or more variables containing at least one equation that is not linear. Any equation that cannot be formed into the equation Ax + By + C = 0 is a nonlinear equation.
• The fifth step is to evaluate the effectiveness of the implemented solutions and make necessary adjustments if needed.
• $\frac{3}{8} =$$0.\overline{375}$
• $\frac{1}{2} =$$0.\overline{4}$
• A function is a relationship between two sets where one set is mapped onto another set by some rule
• A decision tree is a diagram used to illustrate decisions and their possible consequences, helping individuals make informed choices by considering potential outcomes.
• $\frac{1}{6} =$$0.\overline{166...}$
• The sixth step is to monitor the results over time and continuously improve the process as needed.
• Range - The set of all possible output values of a function
• In probability theory, an event space or sample space is the set of all possible outcomes of a random experiment.
• What is the radius of a unit circle?

  1
• What does the equation of a unit circle represent?

  It represents all points at radius 1
• What is the equation of a unit circle?

  $x^2 +$$y^2 =$$1$
• What is the circumference of a unit circle?

  • Circumference = $2\pi r$
  • For unit circle, $r =$$1$
  • Therefore, Circumference = $2\pi$
• How many degrees are in one revolution?

  360°
• What is a radian?

  Measure of angle subtended by arc equal to radius
• How many radians are in one revolution?

  $2\pi$
• How do you convert degrees to radians?

  Multiply by $\frac{\pi}{180°}$
• What is the radian measure of 30°?

  $\frac{\pi}{6} \text{ rad}$
• What is the radian measure of 245°?

  $\frac{49\pi}{36} \text{ rad}$
• What is the radian measure of 810°?

  $\frac{9\pi}{2} \text{ rad}$
• What is the degree measure of $\frac{\pi}{8} \text{ rad}$?


  22.5°
• How long does it take for a searchlight to rotate through 270°?

  13.5 seconds
• What is the degree measure of $5\pi \text{ rad}$?


  900°
• What is a coterminal angle?

  Angles with the same terminal side
• Are 45°, 405°, and -315° coterminal angles?

  Yes
• Are 60°, 780°, and -300° coterminal angles?

  Yes