# Lesson 2

Moving Functions

• Let’s represent vertical and horizontal translations using function notation.

### 2.1: What Happened to the Equation?

Graph each function using technology. Describe how to transform $$f(x)=x^2(x-2)$$ to get to the functions shown here in terms of both the graph and the equation.

1. $$h(x)=x^2(x-2)-5$$
2. $$g(x)=(x-4)^2(x-6)$$

### 2.2: Writing Equations for Vertical Translations

The graph of function $$g$$ is a vertical translation of the graph of polynomial $$f$$.

1. Complete the $$g(x)$$ column of the table.
2. If $$f(0) = \text-0.86$$, what is $$g(0)$$? Explain how you know.
3. Write an equation for $$g(x)$$ in terms of $$f(x)$$ for any input $$x$$.
4. The function $$h$$ can be written in terms of $$f$$ as $$h(x)=f(x)-2.5$$. Complete the $$h(x)$$ column of the table.

$$x$$ $$f(x)$$ $$g(x)$$ $$h(x)=f(x)-2.5$$
-4       0
-3 -5.8
-0.7 0
1.2 -3.3
2 0
5. Sketch the graph of function $$h$$.

6. Write an equation for $$g(x)$$ in terms of $$h(x)$$ for any input $$x$$.

### 2.3: Heating the Kitchen

A bakery kitchen has a thermostat set to $$65^\circ \text{F}$$. Starting at 5:00 a.m., the temperature in the kitchen rises to $$85^\circ \text{F}$$ 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.

1. Sketch a graph of the function $$H$$ that gives the temperature in the kitchen $$H(x)$$, in degrees Fahrenheit, $$x$$ hours after midnight.
2. 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 $$G$$, which gives the temperature in the kitchen as a function of time.
3. Explain what $$H(10.25) = 80$$ means in this situation. Why is this reasonable?
4. If $$H(10.25) = 80$$, then what would the corresponding point on the graph of $$G$$ be? Use function notation to describe the point on the graph of $$G$$.
5. Write an equation for $$G$$ in terms of $$H$$. Explain why your equation makes sense.

Write an equation that defines your piecewise function, $$H$$, algebraically.

### Summary

A pumpkin catapult is used to launch a pumpkin vertically into the air. The function $$h$$ gives the height $$h(t)$$, in feet, of this pumpkin above the ground $$t$$ 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 $$g$$ represent the height $$g(t)$$, in feet, of this pumpkin. How would the graphs of $$h$$ and $$g$$ compare?

Since the height of the second pumpkin is 30 feet greater than the first pumpkin at all times $$t$$, the graph of function $$g$$ is translated up 30 feet from the graph of function $$h$$. For example, the point $$(2,66)$$ on the graph of $$h$$ tells us that $$h(2) = 66$$, so the original pumpkin was 66 feet high after 2 seconds. The new pumpkin would be 30 feet higher than that, so $$g(2) = 96$$. Since all the outputs of $$g$$ are 30 more than the corresponding outputs of $$h$$, we can express $$g(t)$$ in terms of $$h(t)$$ using function notation as $$g(t) = h(t) + 30$$.

Now suppose instead the pumpkin launched 5 seconds later. Let function $$k$$ represent the height $$k(t)$$, in feet of this pumpkin. The graph of $$k$$ is translated right 5 seconds from the graph of $$h$$. We can also say that the output values of $$k$$ are the same as the output values of $$h$$ 5 seconds earlier. For example, $$k(7) = 66$$ and $$h(7-5) = h(2) = 66$$. This means we can express $$k(t)$$ in terms of $$h(t)$$ as $$k(t)=h(t-5).$$