Graphing Transformations

1
Graphing Transformations of the Quadratic Parent
Function
2
Georgia Performance Standard
Graph quadratic functions as transformations of
the function f (x) x2.
3
Vocabulary

• Quadratic Equation Equation in the form y a
  x2 b x c or
• y a (x - h)2 k.

• Parabola The general shape of a quadratic
  equation. It is in the form of a U.

• Vertex The maximum or minimum point of a
  parabola.

• Maximum The highest point of a parabola when it
  opens downward.

• Minimum The lowest point of a parabola when it
  opens upward.

4
"a"
How does the sign of a affect the graph of the
parabola in the equation y a x2?

• On your graphing calculator, do the following
• y x2
• 2. y - x2

5
Opens Up or Down?
parent graph
reflection over x-axis
a lt 0
a gt 0
When the leading coefficient a of x2 is
positive, the graph opens upward.
When the leading coefficient a of x2 is
negative, the graph opens downward.
6
"a"
How does the value of a affect the graph of the
parabola in the equation y a x2?

• On your graphing calculator, do the following
• y 4 x2
• y 1/2 x2 or y 0.5 x2

7
Narrower or Wider?
vertical shrink/horizontal stretch
vertical stretch/horizontal shrink
a gt 1
a lt 1
When the leading coefficient a of x2 is
greater than 1, the graph narrows.
When the leading coefficient a of x2 is less
than 1, the graph widens.
8
Summary for a

• When a is positive, the parabola opens upward.

• When a is negative, the parabola opens downward.

• When a is larger than 1, the graph will be
  narrower (skinnier) than the parent graph of x2.

• When a is smaller than 1, the graph will be wider
  (broader) than the parent graph of x2.

9
"c" and "k"
How does the value of c affect the graph of the
parabola in the equation y x2 c or y x2 - c?

• On your graphing calculator, do the following
• y x2 3
• y x2 - 5

10
Up or Down?
vertical shift downward on the y-axis
vertical shift upward on the y-axis
c gt 0
c lt 0
When the y-intercept of x2 c is positive, the
graph moves up on the y-axis.
When the y-intercept of x2 - c is negative, the
graph moves down on the y-axis.
11
"h"
How does the value of h affect the graph of the
parabola in the equation y (x - h)2 or y (x
h)2?

• On your graphing calculator, do the following
• y (x - 3)2
• y (x 4)2

12
Left or Right?
horizontal shift left on the x-axis
horizontal shift right on the x-axis
y (x h)2 or x h 0 or x h
y (x h)2 or x h 0 or x - h
When the x-intercept is positive h, the graph
moves right on the x-axis.
When the x-intercept is negative h, the graph
moves left on the x-axis.
13
Vertex Summary

• The vertex of y a x2 will be at the origin ?
  (0, 0).

• The vertex of y x2 c will shift up on the
  y-axis ? (0, c).

• The vertex of y x2 - c will shift down on the
  y-axis ? (0, -c).

• The vertex of y (x - h)2 will shift right on
  the x-axis ? (h, 0).

• The vertex of y (x h)2 will shift left on the
  x-axis ? (-h, 0).

14
Practice Problems
Compare the graphs of the following quadratic
equations to each other.
x2 x2 7 -(x 2)2
2x2 -x2 6