MPM2D1
Date: ____________________
Investigation - Transformations of the Graph of y = x2
The graph of a quadratic relation is a ____________________.
You already know two forms for the equation of a quadratic relation.
1. _______________________________
2. _______________________________
There is a third form called vertex form: ______________________________
The simplest quadratic relation is y = x2. In the following exercises, the effects of transformations of the graph of
y = x2 will investigated. Throughout the investigation, students will understand the roles of __________________
in the graph of ___________________.
Part A:
The graph of y = x2 will be used as a comparison for all the
graphs made in the investigation. Begin by graphing y = x2.
You may use a table of values.
x
y
Part B: Comparing y = x2 and y = ax2
1. Using the graphing calculator, construct the following
graphs and sketch the resulting graphs on the axes provided.
Be sure to label each graph so that you can distinguish them.
Choose a different colour for each graph for easier analysis.
a) y = x2
b) y = 2x2
c) y =
1
2
d) y = −
2
x 1
2
e) y = -2x2
x
2
2. Examine the graphs made in #1 and answer the following questions.
a) What is the effect on the graph of y = x2?
b) Without using the graphing calculator, sketch what you
think the graph of y = 3x2 and y = -3x2 will look like.
Verify your sketches using the graphing calculator.
3. Make a summary of your findings by filling in the blanks to complete the following statements.
When graphing y = ax2,
• If a > 0, then the graph of y = ax2 opens ____________________.
• If a < 0, then the graph of y = ax2 opens ____________________.
• If a > 1, then the graph of y = x2 is ____________________ vertically by a factor of ________.
• If 0 < a < 1, then the graph of y = x2 is ____________________ vertically by a factor of ________.
• If a < -1, then the graph of y = x2 is ____________________ vertically by a factor of ________ and
______________________________________________.
• If -1 < a < 0, then the graph of y = x2 is ____________________ vertically by a factor of ________ and
______________________________________________.
Part C: Comparing y = x2 and y = x2 + k
1. Using the graphing calculator, construct the following
graphs and sketch the resulting graphs on the axes provided.
Be sure to label each graph so that you can distinguish them.
Choose a different colour for each graph for easier analysis.
a) y = x2
b) y = x2 + 3
c) y = x2 – 3 d) y = x2 + 5
e) y = x2 – 5
2. Examine the graphs made in #1 and answer the following questions.
a) What is the effect on the graph of y = x2?
b) Without using the graphing calculator, sketch what you
think the graph of y = x2 + 6 and y = x2 – 3 will look like.
Verify your sketches using the graphing calculator.
3. In comparing y = x2 and y = x2 + k,
Part D: Comparing y = x2 and y = (x – h)2
1. Using the graphing calculator, construct the following
graphs and sketch the resulting graphs on the axes provided.
Be sure to label each graph so that you can distinguish them.
Choose a different colour for each graph for easier analysis.
a) y = x2
b) y = (x + 5)2
c) y = (x – 5)2 d) y = (x + 2)2
e) y = (x – 2)2
2. Examine the graphs made in #1 and answer the following questions.
a) What is the effect on the graph of y = x2?
b) Without using the graphing calculator, sketch what you
think the graph of y = (x + 6)2 and y = (x – 1)2 will look
like. Verify your sketches using the graphing calculator.
3. In comparing y = x2 and y = (x – h)2,