Intermediate Math Concepts
5.7 Parent Graphs and Transformations
Name: _______________________
Date: ________________________
Directions: Use the graphing calculator on desmos.com to explore the following functions. Answer the
questions below based on what you find on the calculator.
I. Horizontal and Vertical Shifts
1. Type in the function y = x2 and sketch what you see here:
2. Now type in the functions y = x2 + 2 and y = x2 – 3. Sketch the
new curves on top of y = x2.
3. Type in the function y = x2 and sketch what you see here:
4. Now, type in the functions y = (x + 2)2 and y = (x – 3)2. Sketch the
new curves on top of y = x2.
5. In your own words, describe how adding/subtracting numbers in a quadratic function can change the graph.
Use words like “moves up/down/left/right”.
y = x2 ± d
y = (x ± c)2
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6. In your own words, predict what would happen if you were to type in y = (x + 4)2 + 2.
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7. In your own words, predict what would happen if you were to type in y = -3 + x2.
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II. Horizontal and Vertical Stretches/Shrinks
8. Type in the function y = x2 and sketch what you see here:
9. Now, type in the functions y = 2x2 and y =
1 2
x . Sketch
2
the new curves on top of y = x2.
10. Type in the function y = x2 and sketch what you see here:
11. Now, type in the functions y = (2x)2 and y = (
1 2
x) . Sketch the
2
new curves on top of y = x2.
12. Type in y = 3x2 and y = (3x)2. Do these functions have the same graph? Why or why not?
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13. In your own words, describe how multiplying by numbers larger than or smaller than one in a quadratic
function change the graph. Describe what the new graphs looked like, compared to the original. Use words like
“skinny/wide”.
y = ax2
y = (bx)2
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14. In your own words, predict what would happen if you were to type in y = 4x2.
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15. In your own words, predict what would happen if you were to type in y = (
2 2
x) .
3
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III. Reflections
16. Type in the function y = x2 and sketch what you see here:
17. Now type in y = – x2 and sketch this new curve on top of y = x2.
18. Now type in y = (– x)2. What do you notice?
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19. Why did you get the results you did from #18? Think about what y would be for x = 2 in y = x 2 and
y = (– x)2. What causes both to be y = 4?
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20. Since y = x2 and y = (– x)2 aren’t different, let’s look at a new
Function. Type in the function y = x , and sketch what you see here:
21. Now type in y = –x and sketch this new curve on top of
y= x.
22. In your own words, describe how multiplying by negative numbers can change the graph. Make sure to pay
attention to where the negative multiplier is. Use phrases like “reflects in” and “x or y axis”.
y = – √x
y = √−x
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23. Based on the graphs you sketched in #20, try to sketch what
y = – –x should look like, then use the desmos calculator
to check your work.
IV. Summary
Each of the three “changes” above can be made to any function. Some changes affect functions
differently (ex. The negative in y = (– x)2 did nothing to change y = x2, yet the negative in y = –x did change
the graph of y = x ), but most are very consistent and totally predictable.
Use the desmos calculator to graph the following functions. Write down the changes in the spaces that
are provided. Follow the examples below and use them as a model for your own responses. You do not need to
sketch the graphs on your own. We’ll be doing plenty of that in the days to come.
Example: Graph y =
1
1
3.
and y =
x
x+2
Example: Graph y = x and y = 2 x + 5.
1. Graph y = x2 and y =
1
(x – 4)2.
2
Changes: The second curve has moved Left 2 and Down 3.
Changes: The second curve have moved up 5 and is
skinny (vertical stretch).
Changes: ________________________________________
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2. Graph y = x3 and y = (3x)3 – 1.
Changes: ________________________________________
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3. Graph y =
3
x and y =
3
-x  2
Changes: ________________________________________
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4. Graph y =
1
1
and y = x
x-1
Changes: _________________________________________
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5. Graph y = x and y =
3
x+3  7
4
Changes: _________________________________________
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6. Graph y =
3
x and y = 2 3 x 10
Changes: _________________________________________
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7. Graph y = x2 and y = – 4 – (x + 2)2
Changes: _________________________________________
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