Unit 5, Lesson 2
Moving Functions
Integrated Math 3
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The graph of function is a vertical translation of the graph of polynomial .
Graph of a polynomial function
plane. Horizontal axis scale negative 4 to 2 by 1’s. Vertical axis scale negative 8 to 4 by 2’s. Function f starts down to (negative 4 comma 0), down to (negative 3 comma negative 5 point 8) then up to (negative point 7 comma 0). The function then is down to (1 point 2 comma negative 3 point 3) through a point at (2 comma 0), and increases.
Graph of a polynomial function g, x y plane. Horizontal axis scale negative 4 to 2 by 1’s. Vertical axis scale negative 8 to 4 by 2’s. Function g starts by decreasing through (negative 4 comma 4) down to (negative 3 comma negative 1 point 8) then increasing to (negative point 7 comma 4). The function then decreases to (1 point 2 comma point 7), then up through point (2 comma 4), and continues increases.
Complete the column of the table.
| -4 | 0 | ||
| -3 | -5.8 | ||
| -0.7 | 0 | ||
| 1.2 | -3.3 | ||
| 2 | 0 |
The function can be written in terms of as . Complete the column of the table.
Sketch the graph of function .
A bakery kitchen has a thermostat set to . Starting at 5:00 a.m., the temperature in the kitchen rises to when the ovens and other kitchen equipment are turned on to bake the daily breads and pastries. The ovens are turned off at 10:00 a.m. when the baking finishes.
Sketch a graph of the function that gives the temperature in the kitchen , in degrees Fahrenheit, hours after midnight.
The bakery owner decides to change the shop hours to start and end 2 hours earlier. This means the daily baking schedule will also start and end two hours earlier. Sketch a graph of the new function , which gives the temperature in the kitchen as a function of time.
A pumpkin catapult is used to launch a pumpkin vertically into the air. The function gives the height , in feet, of this pumpkin above the ground seconds after launch.
Now consider what happens if the pumpkin had been launched at the same time, but from a platform 30 feet above the ground. Let function represent the height , in feet, of this pumpkin. How would the graphs of and compare?
Graph of three quadratic functions, origin O. Horizontal axis, time (seconds), scale 0 to 9 by 1’s. Vertical axis, height (feet), scale 0 to 100 by 10’s. Function h starts at the origin, goes through the vertex at (2 comma 66) and back down near (4, 0). Function g starts near (0 comma 30), goes through the vertex at (2 comma 96) and back down near (4 point 5 comma 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 between the vertices of g and h is marked 30 vertically and the distance horizontally between h and k is marked 5.
Since the height of the second pumpkin is 30 feet greater than the first pumpkin at all times , the graph of function is translated up 30 feet from the graph of function . For example, the point on the graph of tells us that , so the original pumpkin was 66 feet high after 2 seconds. The new pumpkin would be 30 feet higher than that, so . Since all the outputs of are 30 more than the corresponding outputs of , we can express in terms of , using function notation as .
Now suppose instead the pumpkin launched 5 seconds later. Let function represent the height , in feet of this pumpkin. The graph of is translated right 5 seconds from the graph of . We can also say that the output values of are the same as the output values of 5 seconds earlier. For example, and . This means we can express in terms of as