Quadratics

Quadratics

1. Graphing

   • Start with y=x^2 and transform all points
   • Plot the vertex and use new step pattern to plot other points
   • Quadratic equations are graphed as parabolas
   • Table of values
     • Can be used to plot points of parabola
     • If 2nd differences are constant, it is a quadratic relationship
       • If 2nd differences are negative, parabola is concave down
       • If 2nd differences are positive, parabola is concave up

2. Roots / Zeroes / X-intercepts

   • They are sometimes called "solutions"
   • They are the x-intercepts of a parabola
   • How to determine number of roots
     • Vertex Form: y = a(x-h)^2 + k
       • If a and k have the same signs, there are no real roots
       • If a and k have different signs, there are two real roots
       • If k is 0, there is one real root (double root)
     • Standard Form: y = ax^2 + bx + c
       • If b^2 - 4ac is equal to 0, there is one real root (double root)
       • If b^2 - 4ac > 0, there are two real roots
       • If b^2 - 4ac < 0, then there are no real roots
   • How to get roots
     • Vertex Form: y = a(x-h)^2 + k
       • Expand and simplify equation to standard form
         • Standard Form
           • Factor into Factored Form (if possible): y = a(x-s)(x-t)
             • The opposite sign of s and t will provide roots
           • Use Quadratic Formula
     • Standard Form

3. Equations

   • Standard form: y = ax^2 + bx + c
     • gives y-intercept, using the value of c
     • Complete the square
       • Vertex Form: y = a(x-h)^2 + k
         • gives vertex (h,k) of parabola
     • Factor equation
       • Factored Form: y = a(x-s)(x-t)
         • Can be expanded and simplified to standard form
   • Factored Form: y = a(x-s)(x-t)
     • Gives the roots of parabola, using the opposite signs of s and t

4. Parabolas

   • Quadratic equations are graphed as parabolas
   • They have a maximum and minimum
     • If parabola is concave up, it has a minimum
       • If the "a" value is positive, parabola is concave up
     • If parabola is concave down, it has a maximum
       • If the "a" value is negative, parabola is concave down
   • They have a vertex
     • The vertex coordinates are represented as (h,k)
     • the k value of the vertex is the optimal value
       • Optimal values are used frequently in word problem
         • When a question asks to maximize or minimize, use the k value of vertex
           • How to find vertex
           • You may also have to find value that gets the max / min, by using the h value of vertex
         • When a question asks you to find break-even / ground point, find x intercept
     • How to find vertex
       • Vertex can be obtained by finding the average of two x coordinates with the same y values
       • If the equation is not in vertex form, expand and simplify it to standard form
         • Complete the square
           • Vertex can be obtained via vertex form, using h and k values
   • They are symmetrical
     • equation of axis of symmetry is determind by x = h

5. Word problems

6. No Frills Parabola

   • Base parabola is y = x^2
   • Transformation of y = a(x-h)^2 + k
     • a value
       • the a value is the stretch factor of a parabola
         • if 0 < |a| < 1
           • The base parabola is vertically compressed by a factor of a
         • if |a| > 1
           • The base parabola is vertically stretched by a factor of a
       • if a < 0
         • The base parabola is reflected on the x-axis, and is concave down
     • h value is the horizontal translation
       • if h > 0, base parabola horizontally translation to the right by h
       • if h < 0, base parabola horizontally translates to the left by h
     • k value is the vertical translation
       • if k > 0, base parabola vertically translates up by k
       • if k < 0, base parabola vertically translates down by k

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