Transforma)ons  of  Parabolas    

Graphs of Parent Functions

f(x) = a(x- h)2 +k

vertex (h, k) •

Horizontal  &  Ver)cal  Shi8s  

Horizontal Shifts:

Vertical Shifts:

Right

Left

Upward

Downward

h(x) = f(x – h)

h(x) = f(x + h)

h(x) = f(x) + k

h(x) = f(x) – k

See  the  Difference  

Vertical Shift: Up two units

Horizontal Shift: Right two units

Reflec)ons  

h(x)  =  -­‐f(x)    h(x)  =  f(-­‐x)    

Reflec4on  about  the  x-­‐axis.      Reflec4on  about  the  y-­‐axis.      

Reflec4on  

General  Transforma)ons  

y  =  a(x  –  h)2  +  k  

a        

c        

d  

If a > 1, the graph moves away from the x-axis (skinnier)

If h > 0, the graph is shifted to the right “h” units.

If k > 0, the graph is shifted up “k” units.

If 0 < a < 1, the graph moves closer to the x-axis (fatter)

If h < 0, the graph is shifted to the left “h” units.

If k < 0, the graph is shifted down “k” units.

Identify the basic function and list all transformations.

g(x) = 2x2 - 4

Basic function: Quadratic Function

The coefficient 2 makes the graph closer to the y-axis. (skinnier)

The -4 shifts the graph down four units.

x

y

Should  we  prac)ce?  Graph using vertex form. y = − 1

2x − 2( )2 +3

Why  don’t  you  try?  Graph y = 2(x + 1)2 – 4 using vertex form. Find y-intercept.

x = 0

y = 2 0+1( )2 − 4 = −2

 More  Prac)ce  

 Write the equation of the parabola below

-1 0 1 2 3 4 5 6

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

•

•(3, 2)

(0, -7)

vertex (3, 2)

po int (0,−7)

y  =  a(x  –  h)2  +  k  

−7 = a 0−3( )2 + 2

−7 = 9a+ 2

a = −1

y  =  -­‐(x  –  3)2  +  2  

-4 -3 -2 -1 0 1

-4

-3

-2

-1

1

2

3

4

5

 Here  is  one  more!!!  

 Write the equation of the parabola below

•

•

(-1,- 2)

(0, 1)

vertex (−1,−2)

po int (0,1)

y  =  a(x  –  h)2  +  k  

1= a 0+1( )2 − 2

1= a− 2

a = 3

y  =  3(x  +  1)2  -­‐  2