Quadratic Equations and Graphing

Quadratic Equations

• Definition: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two.
• General form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• Factoring: Some quadratic equations can be solved by factoring into the product of two binomials.
• Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, used when factoring is not possible.

Graphing

• Cartesian plane: A two-dimensional coordinate system with x-axis and y-axis.
• Graph of a quadratic equation: A parabola that opens upward or downward.
• Key features:
  • Vertex: The lowest or highest point of the parabola.
  • Axis of symmetry: The vertical line that passes through the vertex.
  • x-intercepts: The points where the graph crosses the x-axis.
  • y-intercept: The point where the graph crosses the y-axis.

Surface Area and Volume

Surface Area

• Formula for surface area of a rectangular prism: 2(lw + lh + wh), where l is length, w is width, and h is height.
• Formula for surface area of a cylinder: 2πr(h + r), where r is radius and h is height.

Volume

• Formula for volume of a rectangular prism: lwh, where l is length, w is width, and h is height.
• Formula for volume of a cylinder: πr^2h, where r is radius and h is height.
• Conversion between units: 1 cubic centimeter (cm³) = 1 milliliter (mL)

Quadratic Equations

• A quadratic equation is a polynomial equation with a degree of two, where the highest power of the variable (usually x) is two.
• The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• Some quadratic equations can be solved by factoring into the product of two binomials.
• The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, used when factoring is not possible.

Graphing

• The Cartesian plane is a two-dimensional coordinate system with x-axis and y-axis.
• The graph of a quadratic equation is a parabola that opens upward or downward.
• The key features of a parabola include:
  • The vertex, which is the lowest or highest point of the parabola.
  • The axis of symmetry, which is the vertical line that passes through the vertex.
  • The x-intercepts, which are the points where the graph crosses the x-axis.
  • The y-intercept, which is the point where the graph crosses the y-axis.

Surface Area and Volume

Surface Area

• The formula for the surface area of a rectangular prism is 2(lw + lh + wh), where l is length, w is width, and h is height.
• The formula for the surface area of a cylinder is 2πr(h + r), where r is radius and h is height.

Volume

• The formula for the volume of a rectangular prism is lwh, where l is length, w is width, and h is height.
• The formula for the volume of a cylinder is πr^2h, where r is radius and h is height.
• 1 cubic centimeter (cm³) is equal to 1 milliliter (mL).