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Which three go together? Why do they go together?
A
B
| (hours) | 0.03 | 0.12 | 0.22 | 0.3 | 0.93 | 1.02 | 1.28 | 1.55 | 2.17 | 2.77 | 5.7 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| (degrees Fahrenheit) | 69 | 67.8 | 67.4 | 66.3 | 59.9 | 59.1 | 57.5 | 57.3 | 55 | 52.3 | 47.1 |
A bottle of soda water is left outside on a cold day. The scatter plot shows the temperature , in degrees Fahrenheit, of the bottle hours after it was left outside. Here are 2 functions you can use to model the temperature as a function of time:
Your teacher will give you a card. Take turns describing the transformation of the graph on your card for your partner to draw and drawing the transformed graph from your partner's description.
1. a.Graph of a quadratic function,x y plane. Horizontal and vertical axis, scale negative 6 to 6 by 2’s. The function goes through (negative 2 comma 4) and (negative 1 comma 1), has a vertex at (zero, zero), then rises through (1 comma 1) and (2 comma 4).
1. b.
2. a.Graph of a cubic function, x y plane. Horizontal and vertical axis, scale negative 6 to 6 by 2’s.The function starts near (negative 3 comma negative 7), goes up through (negative 2 comma 0) and down through (negative 1 comma 0) and down through (0 comma negative 2), then up through (1 comma 0) and up near (2 comma 7).
2. b.
3. a.Graph of a polynomial function, x y plane. Horizontal and vertical axis, scale negative 6 to 6 by 2’s. The function starts near (negative 2 comma 7), goes through (negative 2 comma 0) and down near y = negative 2, then back up through (negative 1 comma 0) and grows to (0 comma 4) then down through (1 comma 0) and down near y = negative 2 and then up through (negative 2 comma 0) and keeps growing.
3. b.
4. a.Graph of an exponential function, x y plane. Horizontal and vertical axis, scale negative 6 to 6 by 2’s. The function starts near the x axis, goes through (zero comma 1) and near (1 comma 3) and keeps growing.
4. b.
5. a.
5. b.
6. a.Graph of a quadratic function,x y plane. Horizontal and vertical axis, scale negative 6 to 6 by 2’s. The function goes through (negative 2 comma 4) and (negative 1 comma 1), has a vertex at (zero, zero), then rises through (1 comma 1) and (2 comma 4).
6. b.
The data in the graph show the temperature , in degrees Fahrenheit, of a can of soda hours after it was put into the refrigerator.
Graph of a decreasing exponential function, h T plane. Horizontal axis, scale negative 12 to 18 by 2’s. Vertical axis, scale zero to 60 by 10’s. Function is discrete and has a horizontal asymptote at y = 35.
What if we want to build a function that fits this data set? One way to find a function that fits the data well is to start with a simpler function that has the same general shape as the data when graphed and transform it. What shape does this data form?
Let’s try an exponential decay function. We can get the right shape using a simpler equation like , but the graph doesn't fit where the data is. The graph of the function given by isn't represented by a simple equation, but it does fit the data. (What did multiplying by 45 and adding 36 do to the graph?) In this unit we will learn how to translate, reflect, and stretch graphs to fit data.
Graph of two decreasing exponential functions, h T plane. Horizontal axis, scale negative 12 to 18 by 2’s. Vertical axis, scale zero to 60 by 10’s. Blue function is T = (0 point 7)^h and green function is T = 36 + 45 times(0 point 7)^h.