Quadratic Relationships

Reflection

I learned a lot about quadratics and quadratic formulas throughout this unit. To demonstrate my thorough understanding of this topic , I summarised all my knowledge about Quadratic relationships on this page.

Vertex form

𝑦=𝑎(𝑥−ℎ)² +𝑘

Axis of symmetry:


  • The axis of symmetry divides the parabola into two equal halves
  • The axis of symmetryalways passes through the vertex of the parabola


Vertex of a parabola:

  • The vertex of a parabola is the point where the axis of symmetry and the parabola meet. It is the point where the parabola is at its maximum or minimum value.
Y-intercept:
  • The y-intercept of a parabola is where the graph crosses the y-intercept

Maximum and minimum values:


  • If the parabola opens up, the lowest point is called the vertex (minimum)
  • If the parabola opens down, the vertex is the highest point (maximum)



IDENTIFYING TRANSFORMATIONS OF A PARABOLA



  • The vertex form of a parabola y= a(x-h)² + k
  • The basic parabola has the formula y=x²
  • a- makes the parabola wider or narrower
  • h- is the x-coordinate of the vertex
  • k- is the y-coordinate of the vertex
  • If a>0, the parabola opens up
  • If a<0, the parabola opens down
  • If -1 < a<1, the parabola is vertically compressed
  • If a > 1 or a < -1, the parabola is vertically stretched


Solving:


  • to find the y-intercept, set x=0 and solve for y
  • to solve, set y=0 and solve for x or expand and simplify to get the standard form, then use the quadratic formula

Factored form

𝑦=𝑎(𝑥−𝑟)(𝑥−𝑠)


  • the value of a gives you the shape and direction of opening
  • the value of r and s give you the x-intercepts
  • to find the y-intercept, set x=0 and solve for y
  • Solve using the factors


Types of Factoring:


Greatest Common Factor:


  • the GREATEST COMMON FACTOR (GCF) is the largest integer that divides evenly into each of a given set of numbers.
  • When factoring polynomial expressions, we need to examine both the numerical coefficients and the variables to find the greatest common factor. We look for the greatest common numerical factor, and for the variable with the highest degree of the variable common to each term.
    • We are looking for a whole number factor.
    • To check that we have factored correctly, you can always expand your answer to make sure that we get what we started with because FACTORING IS THE OPPOSITE OPERATION OF EXPANDING.


Simple factoring (a=1):



  • Where x is a variable and a, b, c are constants where a is not 0. A simple trinomial is a quadratic where a = 1
  • Given a quadratic in Standard Form, you can factor to get Factored Form
  • x² +bx+c ( Standard form) = (x+r)(x+s)<-----(factored form)
  • where r+s is b and rs is c,and r and s are integers



Complex factoring:



Special case - Difference of squares:



  • Always look for a common factor first when factoring a trinomial.
  • You can factor a difference of squares as a² -b² = ( a + b) ( a -b ) .
  • We multiply the product of the sum and difference of two terms, therefore
the pattern is ( a+b ) ( a - b )= a² - b².


  • Now we will do the exact opposite (factoring), so the pattern is a² - b²= (a + b)(a - b) .




  • This is called a difference of squares. Both a and b must be perfect squares (if you take the square root, it is a whole number). Some perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...




Special case – Perfect square:



The trinomial that results from squaring a binomial is called a perfect square trinomial. Perfect square trinomials can be factored using the patterns from squaring binomials. You can factor a perfect square trinomial as


  • a² +2ab+b² = (a+b)²

    a²-2ab+b² = (a-b)²



    Middle term should be twice the product of the square roots of the first and last terms.


    2(√a x √b ) = 2ab




    In a perfect square trinomial, the first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms.

Standard form

𝑦=𝑎𝑥² +𝑏𝑥+𝑐


  • the value of a gives you the shape and direction of opening
  • the value of c is the y-intercept
  • Solve using the quadratic formula, to get the x-intercepts
  • MAX or MIN? Complete the square to get vertex form



  • The quantity b²-4ac in the quadratic formula is called the discriminant
  • If b²-4ac then we will get two real solutions to the quadratic equation.
  • If b²-4ac then we will get a double root to the quadratic equation.
  • If b²-4ac then we will get two complex solutions to the quadratic equation.