Quadratic Equations Adapted from: Vertex Formula Quadratic Equations Adapted from:

Vertex Formula: Y = a( x - h )2 + k A=Second constant difference divided by 2 H=horizontal shift K=Vertical shift

Vertex The vertex is the lowest or highest point on our parabola. SO the vertex is also the maximum or minimum of the function It is where the axis of symmetry intersects the parabola The vertex is (h,k)

Axis of Symmetry The axis of symmetry is a vertical line that passes through the vertex You could think of it as the “mirror” that makes both sides of the parabola mirror images The axis of symmetry goes through the center of the parabola The axis of symmetry is x=h

Transformations of Quadratic Graphs Transformation means CHANGE (think Transformers!) Types of transformations: Reflections Vertical Stretch Horizontal Stretch Vertical or Horizontal Translations To describe the transformations of the parabola, describe the changes from the original

Reflection Reflection refers to the parabola reflecting, or inverting, around the x- axis This is determined by the sign on a If a is POSITIVE, the parabola will open UP (like a smiley face  ) If a is NEGATIVE, the parabola will open DOWN (like a sad face  )

Vertical Stretch Vertical Stretch makes your parabola NARROWER (think of the sticky tack example from class) If the absolute value of a is greater than 1, the parabola will get narrower or skinnier Vertical stretch multiplies all of the y values of the function by the same factor greater than 1

Horizontal Stretch Horizontal Stretch is when the graph widens or gets fatter When the absolute value of a is less than 1, the graph will get WIDER Horizontal stretch decreases all of the y values of the function by the same factor less than 1.

Vertical Translations Vertical translations occur when the parabola is moved up or down without changing the shape of the parabola (vertical shift) In the vertex formula, k indicates vertical translation If k is POSITIVE, the parabola moves UP If k is NEGATIVE, the parabola moves DOWN

Horizontal Translations Horizontal translations occur when the parabola moves right or left In vertex formula, h represents horizontal shift This is the tricky part - h is the opposite of what you would expect!! If h is positive, the parabola moves LEFT (yes, towards the negative x values!) If h is NEGATIVE, the parabola moves RIGHT towards the larger positive x values WHY??

Why is h Opposite? Y = a( x - h )2 + k In our vertex formula, notice there is already a minus sign before h! Y = a( x - h )2 + k So, if you substitute h with a positive number, say, 2, the minus sign stays there!! So, you would have something like: y = a ( x – 2 )2 And if you substitute h with a negative number, like -3, the minus sign is now next to a negative sign. *subtracting a negative number is the equivalent to addition! So it would become: y = a ( x + 3 )2

What Now? Use these links for extra help and demonstrations with vertex form  http://www.mathwarehouse.com/geometry/parabola/axis-of- symmetry.php http://www.mathwarehouse.com/geometry/parabola/vertex-of-a- parabola.php http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/geom etry/graphtranslationiandstretch.html http://www.purplemath.com/modules/fcntrans2.htm