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?

• 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