Quadratic relations

Quadratic relations

1. Quadratic Relations (Parabolas)

   • Vertex
   • Axis of Symmetry
   • Direction of opening
   • Zero/ Roots/ x-intercepts
   • Optimal values
   • y-intercept

2. Transformation of Quadratics

   • Vertical translations
     • if k > 0, vertical translation/ shift k units up
     • if k < 0, vertical translation/ shift k units down
   • Horizontal translation
     • if h < 0, horizontal translation/ shift h units to the left
     • if h > 0, horizontal translation/ shift h units to the right
   • Vertical Stretch
     • if a > 1, vertical stretch by a factor of a
   • Vertical Compression
     • if 0 < a < 1, vertical compression by a factor of a
   • Reflection in the x-axis
     • a < 0, reflection in the x-axis

3. Form of a Quadratic

   • Standard form
     • y = ax² + bx + c
   • Vertex form
     • y = a(x - h)² + k
       • a > 0, opens up
         • has min
       • a < 0, opens down
         • has max
       • vertex
         • (h , k)
   • Factor form
     • y = a(x - r)(x - s)
       • x-intercepts are - (r) and - (s)
       • AOS
         • (r + s) / 2
           • Vx
       • Vy
         • The y-coordinate of the vertex is found by subtituting the vx into the equation

4. Word Problems

   • Pythagorean Theorem
     • a² + b² = c²
   • Consecutive Integers
     • 1, 2, 3, 4, 5
   • Revenue
     • Revenue = (Quantity)(Price)

5. Quadratic Equations

   • Factoring
     • y = 0
     • writen in the form ax² + bx + c = 0
     • factor fully
     • determine the value of x that make each factor equal to zero
   • Completing the Square
     • (a + b)²
       • a² + 2ab +b²
     • (a - b)²
       • a² - 2ab + b²
     • Group the first two terms (Only if a ≠ 1)
     • factor the a from both variable (Only if a ≠ 1)
     • take the coefficient of x, divide it by 2, and square it
     • add and subtract by this quantity in order to complete the square
     • factor the prefect square trinomial (a² ± 2ab + b²)
     • move the subtracted value outside of the brackets by multiplying it by the a factor
     • simplify
   • Quadratic formula
     • x = (-b ± √b² - 4ac)/ 2
       • the quadratic equation MUST be in standard form (ax² + bx + c)
     • discriminant
       • b² - 4ac > 0
         • has 2 real roots
       • b² - 4ac = 0
         • has 1 real roots
       • b² - 4ac < 0
         • has NO real roots

6. Multiplying Polynomials

   • Standard form
     • y = ax² + bx + c
   • Vertex form
     • y = a (x - h)² + k
   • Factored form
     • y = a (x - r)(x - s)
   • FOIL
     • First
     • Outside
     • Inside
     • Last
     • how to do
   • Polynomials
     • Binomials
       • Polinomials with 2 terms
     • Trinomials
       • Polinomials with 3 terms

7. Special products

   • Perferct Square Trinomials
     • (a + b)²
       • a² + 2ab + b²
     • (a - b)²
       • a² - 2ab + b²
   • Difference of squares
     • (a + b)(a - b)
       • a² - b²

8. Common factors

   • Factor: to express a number as a product of two or more numbers
     • Factors of 12 = 1x12, 2x6, 3x4
     • Factors of 16 = 1x16, 2x8, 4x4
   • Greatest Common Factor (GCF): the largest number that factors/ divides evenly in 2 or more numbers
     • GCF of 12 and 16 is 4
   • Expand (Distributive Property/ FOIL)
     • 2 (3x + 4)
       • 6x + 8
     • (x - 1)(3x + 2)
       • 3x² + 2x - 3x -2
         • 3x² - x - 2

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