# Lesson 3

More Movement

• Let’s translate graphs vertically and horizontally to match situations.

### 3.1: Moving a Graph

How can we translate the graph of $$A$$ to match one of the other graphs?

### 3.2: New Hours for the Kitchen

Remember the bakery with the thermostat set to $$65^\circ \text{F}$$? At 5:00 a.m., the temperature in the kitchen rises to $$85^\circ \text{F}$$ 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.

1. Andre thinks, “When the bakery starts baking 2 hours earlier, that means I need to subtract 2 from $$x$$ and that $$G(x)=H(x-2)$$.” How could you help Andre understand the error in his thinking?
2. The graph of $$F$$ shows the temperatures after a change to the thermostat settings. What did the owner do?
3. Write an expression for $$F$$ in terms of the original baking schedule, $$H$$.

### 3.3: Thawing Meat

A piece of meat is taken out of the freezer to thaw. The data points show its temperature $$M$$, in degrees Fahrenheit, $$t$$ hours after it was taken out. The graph $$M=G(t)$$, where $$G(t) = \text-62(0.85)^t$$, models the shape of the data but is in the wrong position.

$$t$$ $$M$$
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

Jada thinks changing the equation to $$J(t)=\text-62(0.85)^t + 75.1$$ makes a good model for the data. Noah thinks $$N(t) = \text-62(0.85)^{(t+1)}+68$$ is better.

1. Without graphing, describe how Jada and Noah each transformed the graph of $$G$$ to make their new functions to fit the data.
2. Use technology to graph the data, $$J$$ and $$N$$, on the same axes.
3. Whose function do you think best fits the data? Be prepared to explain your reasoning.

Elena excludes the first data point and chooses a linear model, $$E(t)=21.32+5.06t$$, to fit the remaining data.

1. How well does Elena’s model fit the data?
2. Is Elena’s idea to exclude the first data point a good one? Explain your reasoning.

### Summary

Remember the pumpkin catapult? The function $$h$$ gives the height $$h(t)$$, in feet, of the pumpkin above the ground $$t$$ seconds after launch. Now suppose $$k$$ represents the height $$k(t)$$, in feet, of the pumpkin if it were launched 5 seconds later. If we graph $$k$$ and $$h$$ on the same axes they looks identical, but the graph of $$k$$ is translated 5 units to the right of the graph of $$g$$.

Since we know the pumpkin's height $$k(t)$$ at time $$t$$ is the same as the height $$h(t)$$ of the original pumpkin at time $$t-5$$, we can write $$k$$ in terms of $$h$$ as $$k(t)=h(t-5)$$.

Suppose there was a third function, $$j$$, where $$j(t)=h(t+4)$$. Even without graphing $$j$$, we know that the graph reaches a maximum height of 66 feet. To evaluate $$j(t)$$ we evaluate $$h$$ at the input $$t+4$$, which is zero when $$t = \text-4$$. So the graph of $$j$$ is translated 4 seconds to the left of the graph of $$h$$. This means that $$j(t)$$ is the height, in feet, of a pumpkin launched from the catapult 4 seconds earlier.