quadratic

quadratic

1. Multiplying Polynomials

   • Standard form
     • y=ax²+bx+c,a≠0
   • factored form
     • y=a(x-r)(x-s)
   • Vertex form
     • y=a(x-h)+k

2. Investigage Transformations of Quadratics

   • y=x²±k
     • if k＞0, vertical translate/shift k units up
     • if k＜0, vertical translate /shift k units down
   • y=(x±h）²
     • if h＞0, horizontal translate/ shift h units to the right.
     • if h＜0, horizontal translate/ shift h units to the left.
   • y= ax²
     • if a＞1, vertical stretch by a factor of "a".
     • if 0＜a＜1, vertical compression by a factor of "a".
     • if a＜0, reflection in the x-axis.

3. Graph

   • graph vertex form: y=a(x-h)²+k.
     • a＞0,opens up (has min)
     • a＜0, opens down (has max)
     • horizontal translation axis of symmetry is x=h.
     • Vertical translation min/max value is y=k.
     • 1. Plot the vertex (h,k).
     • 2. Use a and the pattern to count over and up/down to get 4 more points.
     • 3. Draw a smooth curve
   • graph factor form: y= a(x-r)(x-s).
     • Represents a quadratic equation in factored form.
     • The x-intercepts/zeros are located at (r,0) and (s,0).
     • The axis of symmetry is located at: x=(r+s)/2
     • The x-coordinate of the vertex is value of the axis of symmetry.
     • The y-coordinate of the vertex is found by substituting the x-coordinate into eqyation.

4. Zero and negative exponents

   • product rule:a^x * a^y=a^x+y
   • quotient rule:a^x ÷ a^y =a^x-y
   • power to a power rule:(a^x)^y =a^xy
   • a^0=1

5. Special products

   • Perfect Square Trinomials
     • (a+b)(a-b)=a²-b²
   • Difference of Squares
     • (a+b)² =a²+2ab+b²
     • （a-b)²=a²-2ab+b²

6. Factor

   • common factors
     • Greatest Common Factor(GCF):the largest number that factor/divides evenly in 2 or more numbers
     • Common factoring is the opposite of the distributive property.
     • 1. Find the GCF of all the terms.
     • 2. Take out the GCF of the polynoimial (factor it) by dividing each term by the GCF.
   • Factoring Simple Trinomials
     • Not every trinomial is factorable (r,s don't exist)
     • If a quadratic expression of the form x²+bx+c is factorable, it can be factored into (x+r) and (x+s), where b=r+s and c=r*s.
     • How to factor: Find two numbers, r and s, that add up to b and multiply to c.
   • Factoring Complex Trinomials
     • Factoring by decomposition
       • To decompose the middle term, we need to find two numbers that add up to b and multiple to ac.
     • not all quadratic expressions of the form ax²+bx+c can be factor.
   • Special Cases
     • Factored by Difference Of Squares
       • a²-b²=(a+b)(a-b)
     • Factored by perfect Square Trinomial
       • (a+b)² =a²+2ab+b²
       • a-b)²=a²-2ab+b²

7. Solve Quadratic Equation

   • Factoring
     • The quadratic relation in factored form y= a(x-r)(x-s) where r and s are the zeros.
     • To solve a Quadratic Equation:
       • 1. Write in the form ax²+bx+c=0
       • 2.Factor fully
       • 3.Determine the value of x that makes each factor equal to 0.
   • The Quadratic Formula
     • The quadratic quadratic MUST be in standard form y=ax²+bx+c before it can be solved using the quadratic formula.
     • The discriminant b²-4ac determines how many solutions the quadratic equation has:
     • 1. If b²-4ac＞0, the n the quadratic equation has 2 real roots.
     • 2. If b²-4ac＝0, then the quadratic equation has 1 ral root.
     • 3, If b²-4ac＜0, then the quadratic equation has no real roots,

8. Graphing Quadratics

   • x-intercepts
     • The axis symmetry is located at:x=r+s/2
     • The x-coordinate of the vertex is the value of the Alos.
     • The y-coordinate of the vertex is found by substituting the x into the equation.

9. quadratic relations

   • graphs: parabola
     • These are graphs of quadratic relations. The graph is called a parabola.
     • The vertex of a parabola is either the minimum point(open up) or maximum point(open down)
     • A vertex line of symmetry witch goes through the vertex is called the axis of symmetry (x=...)
     • The x-intercept of a parabola are called its zeros or roots.
   • equation: y=ax²+bx+c
     • THe equation of a quadration relation can be writte nin the form y=ax²+bx+c, where a,b and c are constants and a≠0.

10. Completing the Square

    • The process of completing the square allows you to change quadratic relation from Standard Form to vertex form.
    • 1. Group the first two terms.
    • 2. Factor the a from both variable terms.
    • 3. Take the coefficient of x, divide it by 2 and square it.
    • 4. Add and substract by this quantity in order to complete the square.
    • 5. Factor the perfect square trinomial.
    • 6. More the substracted value outside of the brackets by multiplying it by the a factor.
    • 7.Simplify

11. Maxima and Minma Word Problems

    • 1. Revenue= quantity*price
    • 2. "let" statement and end with a "∴" statement.
    • restrictions and units.

12. Solve Problem Using Quadratic Equations

    • 1. Define your variables("let" statement)
    • 2. Write an equation to model the situation.
    • 3. Simplify the equation if necessary.
    • 4, Solve the equation using the appropriate method(factor/complete the square/quadratic formula).
    • 5. Reject a solution if necessary, providing an appropriate reason.
    • 6. State therefore statement, answering the original question.