Autograph Quadratics Investigation
1. Open My Computer and double click on the Student Drive (X). Open the folder labelled Service and
open the Autograph file called Quad Transformations.
On the screen, you will see a graph of the quadratic function y  x 2 .
Give the coordinates of the vertex:
________
Starting from the vertex, if you:
move right 1 space, move up ________________; move left 1 space, move up ________________
move right 2 spaces, move up ________________; move left 2 spaces, move up ________________
move right 3 spaces, move up ________________; move left 3 spaces, move up ________________
2. Now you are going to compare graphs of other quadratic functions to y  x 2 . Notice after the graph it
says Equation 2: a( y  k )  ( x  h) 2 . This is the transformational form of a quadratic function. You are
going to investigate what happens when the values of a, k and h are changed by comparing the graphs to
y  x2 .
On the top menu bar, click on the Constant
Controller button.
. (drag the window
so it does not block your graph).
(a) Select the h constant first. Use the up arrow to increase the value of h. What happens to the graph?
Use the down arrow to decrease the value of h. What happens to the graph?
Set h = 5. What is the vertex?
________
Set h = -6. What is the vertex?
________
(b) Reset the h constant to 0. Select the k constant. Use the up arrow to increase the value of k. What
happens to the graph?
Use the down arrow to decrease the value of k. What happens to the graph?
Set k = 4. What is the vertex?
________
Set h = 4 and k = -3. What is the vertex?
Set k = -5. What is the vertex?
________
________
What would the vertex of the graph y  8  ( x  5) 2 be?
________
(c) Reset h and k to 0. Select the a constant. Use the up arrow to increase the value of a. What happens
to the graph?
Use the down arrow to decrease the value of a, but do not go below 0! What happens to the graph when
a is less than one?
Keep decreasing the value of a, so that it becomes negative. What happens to the graph?
Set a = ½ (use 0.5). Compare the graph to y  x 2 :
# Spaces moving left or right
# Spaces moving up
from vertex
( y  x2 )
1
1
2
4
# Spaces moving up
1
( y  x2 )
2
Set a = 2. Compare the graph to y  x 2 :
# Spaces moving left or right
# Spaces moving up
from vertex
( y  x2 )
1
1
2
4
3
9
# Spaces moving up
( 2y  x2 )
Set a = 3. Compare the graph to y  x 2 :
# Spaces moving left or right
# Spaces moving up
from vertex
( y  x2 )
1
1
2
4
3
9
# Spaces moving up
( 3y  x 2 )
3. Draw graphs for the following functions. Label the vertex of each.
(a) y  2  ( x  4) 2
10
(b) y  8  ( x  5) 2
y
10
10
y
8
8
8
6
6
6
4
4
4
2
2
x
–10 –8 –6 –4 –2
–2
(d)
(c)  y  ( x  6) 2
2
4
6
8
10
2
x
–10 –8 –6 –4 –2
–2
2
4
6
8
10
x
–10 –8 –6 –4 –2
–2
–4
–4
–4
–6
–6
–6
–8
–8
–8
–10
–10
–10
1
( y  10)  x 2
3
10
10
10
y
8
8
8
6
6
6
4
4
4
2
2
2
4
6
8
10
4
6
8
y
x
–10 –8 –6 –4 –2
–2
10
2
x
–10 –8 –6 –4 –2
–2
2
1
(f)  ( y  8)  ( x  5) 2
2
(e) 2( y  3)  ( x  4) 2
y
y
2
4
6
8
10
x
–10 –8 –6 –4 –2
–2
–4
–4
–4
–6
–6
–6
–8
–8
–8
–10
–10
–10
2
4
6
8
10