Mathematics Quarter 1 - Grade 9

Rational Algebraic Equations

• Equations can be transformed into quadratic forms for easier solving.
• Example equation options provided:
  • ( m + \frac{1}{2} + m + \frac{2}{2} = 7 )
  • ( 2m - \frac{1}{3} + 1 - \frac{2}{4} = 3 )
  • ( \frac{3m}{4} + \frac{m + 1}{4} = 5 )
  • ( \frac{m}{3} + \frac{m + 1}{m} - \frac{2}{m + 2} = 7 )

Solutions to Specific Equations

• The solutions for the equation ((x + 3)^2 + 2(x + 3) - 8 = 0) are:
  • Possible answer options include:
    • (-7, -1)
    • (7, 1)
    • (3, 8)
    • (-3, -8)

Quadratic Transformations

• The equation ( \frac{2x - 1}{x + 1} = \frac{x + 1}{x - 1} ) leads to quadratic forms.
• Possible quadratic transformations include:
  • ( x^2 - 5x = 0 )
  • ( x^2 - 5x + 1 = 0 )
  • ( x^2 + 5x = 0 )
  • ( x^2 + 5x + 1 = 0 )

Solutions to Quadratic Equations

• For the quadratic transformation from the previous equation, potential solutions include:
  • ( 0, -5 )
  • ( 0, 5 )
  • ( 3 \pm \frac{\sqrt{6}}{3} )
  • ( -3 \pm \frac{\sqrt{6}}{3} )

Projectile Motion Problem

• An object is thrown upward from a 300 ft tall building, modeled by the equation:
  • ( h = -4t^2 + 40t + 300 )
• To find the height at 5 seconds after release, evaluate the equation for ( t = 5 ):
  • Possible height options include:
    • 160 ft
    • 400 ft
    • 600 ft
    • 1000 ft

Time to Hit the Ground

• To find the time it takes for the object to hit the ground, solve for ( t ) when ( h = 0 ):
  • Possible time options include:
    • 5 seconds
    • 10 seconds
    • 12 seconds
    • 15 seconds

Quadratic Functions and Graphs

• The graph of quadratic functions is known as a parabola.
• Quadratic functions are characterized by an equation in which the highest power of the variable is 2.

Understanding Quadratic Equations

• The equation ( y + 2x^2 = 2x(x + 1) ) represents a quadratic function because the highest degree of x, after simplifying, is 2.
• Quadratic functions cannot be expressed in the form ( y = mx + b ) as this format depicts linear functions.

Analyzing Data Tables

• A table of values with outcomes where all y values are positive does not necessarily indicate a quadratic relationship.
• To confirm if a table represents a quadratic function, one must check if the first differences (differences between consecutive y values) are equal or if the second differences are equal.

Completing the Table of Values

• For the quadratic equation ( y = -2x^2 + 4x + 7 ), specific values of ( m ) and ( n ) need to be calculated to fill in the table correctly.

Characteristics of Quadratic Functions

• A quadratic function is generally expressed as ( y = ax^2 + bx + c ), where ( a > 0 ) indicates the parabola opens upwards.
• The vertex form of a quadratic function is given as ( y = a(x-h)^2 + k ), where ( (h, k) ) represents the vertex of the parabola.

Visualization of Quadratics

• Graphs describing a quadratic function with ( a > 0 ) depict a parabola that opens upwards.