Unit 5, Lesson 3
More Movement
Integrated Math 3
3.1
Warm-up
How can we translate the graph of A to match one of the other graphs?
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How can we translate the graph of A to match one of the other graphs?
Remember the bakery with the thermostat set to ? At 5:00 a.m., the temperature in the kitchen rises to due to the ovens and other kitchen equipment being used until they are turned off at 10:00 a.m. When the owner decided to open 2 hours earlier, the baking schedule changed to match.
Graph of function
. Horizontal axis, hours after midnight, scale 0 to 24 by 2’s. Vertical axis, degrees Fahrenheit, scale 50 to 100 by 10’s. Function starts at (0 comma 70) and is horizontal to (6 comma 70), then rises to (7 comma 90) and is horizontal to (11 comma 90). The function then drops to (12 comma 70) and is horizontal to (24 comma 70).
A piece of meat is taken out of the freezer to thaw. The data points show its temperature , in degrees Fahrenheit, hours after it was taken out. The graph , where , models the shape of the data but is in the wrong position.
| 0 | 13.1 |
| 0.41 | 22.9 |
| 1.84 | 29 |
| 2.37 | 36.1 |
| 2.95 | 36.8 |
| 3.53 | 38.8 |
| 3.74 | 40 |
| 4.17 | 42.2 |
| 4.92 | 45.8 |
Coordinate plane. Horizontal axis, t, hours, from 0 to 10 by 5’s. Vertical axis, M, degrees Fahrenheit, from negative 60 to 60 by 20’s. 9 data points plotted at 0 comma 13 point 1, 0 point 41 comma 22 point 9, 1 point 84 comma 29, 2 point 37 comma 36 point 1, 2 point 95 comma 36 point 8, 3 point 53 comma 38 point 8, 3 point 74 comma 40, 4 point 17 comma 42 point 2, 4 point 92 comma 45 point 8. Function moving upwards and to the right passing through negative 62 comma 0, 5 comma negative 27, and 10 comma negative 12.
Jada thinks changing the equation to makes a good model for the data. Noah thinks is better.
Remember the pumpkin catapult? The function gives the height , in feet, of the pumpkin above the ground seconds after launch.
Now suppose represents the height , in feet, of the pumpkin if it were launched 5 seconds later. If we graph and on the same axes they looks identical, but the graph of is translated 5 units to the right of the graph of .
Since we know the pumpkin's height at time is the same as the height of the original pumpkin at time , we can write in terms of as .
Graph of two quadratic functions, origin O. Horizontal axis, time (seconds), scale 0 to 9 by 1’s. Vertical axis, height (feet), scale 0 to 90 by 10’s. Function h starts at the origin, goes through the vertex at (2 comma 66) and back down near (4, 0). Function k starts at (5 comma 0), goes through the vertex at (7 comma 66) and back down near (9 comma 0). The distance horizontally between the vertices of h and k is 5.
Suppose there was a third function, , where . Even without graphing , we know that the graph reaches a maximum height of 66 feet. To evaluate , we evaluate at the input , which is zero when . So the graph of is translated 4 seconds to the left of the graph of . This means that is the height, in feet, of a pumpkin launched from the catapult 4 seconds earlier.