2nd Quarter - [1st Sem] Pre-Calculus Reviewer

Module 3: Ellipses

Definition of Ellipse

• An ellipse is a set of all points on a plane where the sum of the distances from two fixed points (foci) is constant.

Key Components

• Foci: Two fixed points within the ellipse.

• Major Axis: The longer axis containing the foci, vertices, and center.

• Minor Axis: The shorter axis that is a perpendicular bisector of the major axis.

• Vertices: Endpoints of the major axis (±a).

• Co-vertices: Endpoints of the minor axis (±b).

• Center: Intersection of the axes of symmetry.

• Latus Rectum: A chord passing through a focus, perpendicular to the major axis.

• Directrix: A line parallel to the minor axis, equidistant from the vertex as the focus.

Forms of an Ellipse

• General Form:

  • Center at origin:[ A x^2 + B y^2 + C = 0 ]

  • Center at (h, k):[ A x^2 + B y^2 + C x + D y + E = 0 ]

• Standard Form:

  • Center (h, k):[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 ]

Module 3 Continued: Graphing an Ellipse

Steps in Graphing

1. Locate center using (h, k).

2. Determine ellipse type and its major axis using a².

3. Solve for a, b, and c.

4. Locate coordinates of vertices, co-vertices, and foci.

5. Connect vertices and co-vertices to outline the ellipse.

6. Draw latus rectum through foci.

7. Sketch directrices parallel to the minor axis.

8. Label all parts with coordinates or equations.

Module 3 Continued: Finding Equations of an Ellipse

Key Concepts

1. Requirements to write the equation:

   • a. Coordinates (h, k)

   • b. Ellipse type (horizontal or vertical)

   • c. Length of major axis for a²

   • d. Length of minor axis for b²

2. Convert standard form to general form as needed.

Module 4: Hyperbola

Definition of Hyperbola

• A hyperbola is a smooth curve defined geometrically by the foci such that the difference of distances from any point on the hyperbola to the foci is constant.

Forms of Hyperbola

• General Form:

  • [ A x^2 - B y^2 + E y + F = 0 ]

  • [ B y^2 - A x^2 + E y + F = 0 ]

• Standard Form:

  • Centered at (0, 0) or at (h, k).

Module 4 Continued: Sketching a Hyperbola

Key Components

• Transverse Axis: Line through the foci intersecting at vertices.

• Center: Midpoint of the transverse axis.

• Conjugate Axis: Perpendicular to the transverse axis.

• Distances:

  • Vertex to center = a

  • Focus to center = c

  • Relationship: c² = a² + b².

• Asymptotes: Straight lines through the center illustrating the behavior of the hyperbola.

Module 4 Continued: Finding Hyperbola Equations

Steps

1. Determine a and c values.

2. Identify transverse axis direction (horizontal/vertical).

3. Compute b using c² = a² + b².

4. Formulate standard equation based on the transverse axis.

5. Substitute a and b into the equation.

Module 6: Unit Circle

Concept of Trigonometry

• Trigonometry deals with the ratios of sides in right triangles and angles.

Key Angle Concepts

• Angle Formation: An angle is two rays meeting at a vertex.

• Standard Position: Angle with vertex at the origin and initial ray on the positive x-axis.

• Co-terminal Angles: Angles sharing the same terminal side; calculated by adding/subtracting multiples of 360°.

• Reference Angles: Acute angle between terminal side and x-axis.

Module 6 Continued: Radian and Degree Measures

Definitions and Conversion

• 1 radian = arc length = r; 2π radians in a circle (360°).

• Conversions:

  • Degrees to Radians: multiply by π/180

  • Radians to Degrees: multiply by 180/π.

Module 6 Continued: Understanding the Unit Circle

Characteristics

• Unit Circle: Radius of 1.

• Central Angle: Vertex at the center, sides are radii.

• Radian Measure: Ratio of arc length to radius ([ \theta = \frac{s}{r} ]).

Module 6 Continued: Circular Functions

Definitions

• Six circular functions defined by coordinates of the terminal point:

  • sin(θ) = y, cos(θ) = x, tan(θ) = y/x (x ≠ 0), etc.

Module 6 Continued: Graphs of Circular Functions

Characteristics

• Periodic Functions: Repeat patterns at regular intervals.

• Sine and Cosine: Period = 2π, amplitude = 1.

• Tangent and Cotangent: Different period (π) and undefined amplitudes.

Tips for Applied Problems

• Utilize the six trigonometric ratios from right triangles to solve for missing lengths and angles.