Ch. 5 Notes Page 31P31 5.3: Transforming Parabolas

“I am only one, but I am one. I cannot do everything, but I can do something. And I will not

let what I cannot do interfere with what I can do.” – Edward Everett Hale

Vertex Form of a Quadratic

Vertex Form of a Quadratic Equation:

f(x) = a(x – h)2 + k

Parent Function: y = x2 Sketch:

Vertical flip if a is negative,

vertical stretch (a > 1) or shrink (a < 1)

Horizontal translation

(opposite of what you see!)

Vertical translation

*The vertex of the parabola is (h, k) and the axis of symmetry is x = h.

1. Vertex (horiz. and vert. translation)

2. Axis of symmetry

3. Table

• Point

• Vertex

• Reflection

4. Ask:

• Correct reflection?

• Correct stretch or shrink?

Graphing Equations in Vertex Form

x y

3)2(21 2 xy

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Try this one…

4)1(2)( 2 xxf

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

1. Vertex (horiz. and vert. translation)

2. Axis of symmetry

3. Table

• Point

• Vertex

• Reflection

4. Ask:

• Correct reflection?

• Correct stretch or shrink?

x y

Writing the equation.

Write the equation for the following parabola in vertex form:

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Need:

• Vertex (h, k)

• Another point (x, y)

(so we’re solving for a)

y = a(x – h)2 + k

Do the transformations seem correct?

Distance (?)

Real Life!

The Verrazano-Narrows Bridge in New York has the longest span of any suspension bridge in the US. The curve of a suspension cable on this bridge can be modeled by the function

y = 0.0001432(x - 2130)2,

where x and y are measured in feet. The origin of the graph is at the base of one of the two towers supporting the cable. How far apart are the towers? How high are they?

Height (?)

Changing an Equation from Standard to Vertex Form

Write the equation in vertex form. 5123 2 xxy

1. Find the x-coordinate of the vertex (h):

2. Find the y-coordinate of the vertex (k):

3. Substitute a, h, and k into vertex form:

ab

2

khxay 2)(

5.3: Transforming Parabolas

HW#26 5.3: P255 #1, 4, 9, 14, 16, 19, 25-27, 34, 42bcd, 47, 53