# Lesson 2

Moving Functions

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

### Problem 1

The thermostat in an empty apartment is set to $$65^\circ \text{F}$$ from 4:00 a.m. to 5:00 p.m. and to $$50^\circ \text{F}$$ from 5:00 p.m. until 4:00 a.m. Here is a graph of the function $$H$$ that gives the temperature $$H(x)$$ in degrees Fahrenheit in the apartment $$x$$ hours after midnight.

1. The owner of the apartment decides to change to a new schedule and they set the thermostat to change 3 hours later in the morning and the evening. On the same axes, sketch a graph of the new function, $$G$$, giving the temperature as a function of time.
2. Explain what $$H(6.5) = 65$$ means in this context. Why is this a reasonable value for the function?
3. If $$H(6.5) = 65$$, then what is the corresponding point on the graph of $$G$$? Use function notation to describe the point on the graph of $$G$$.
4. Write an expression for $$G$$ in terms of $$H$$.

### Problem 2

A pumpkin pie recipe says to bake the pie at $$425^\circ \text{F}$$ for 15 minutes, and then to adjust the temperature down to $$350^\circ \text{F}$$ for 45 additional minutes. The function $$P$$ gives the oven temperature setting $$P(t)$$, in degrees Fahrenheit, $$t$$ minutes after the pie is placed in the oven.

1. Explain what $$P(30)=350$$ means in this context.
2. Diego discovers that the temperature inside the oven is always 25 degrees warmer than the oven’s temperature setting. The function $$B$$ gives the actual temperature of Diego’s oven. If $$P(30)=350$$, then what is the corresponding point on the function $$B$$?
3. Write an expression for $$B$$ in terms of $$P$$.

### Problem 3

Here is the graph of $$y = f(x)$$ for a function $$f$$.

1. On the same axes, sketch a graph of $$g(x) = f(x) + 2$$.
2. On the same axes, sketch a graph of $$h(x) = f(x+2)$$.
3. How do the graphs of $$g$$ and $$h$$ compare to $$f$$?

### Problem 4

The graph shows the height of a tennis ball $$t$$ seconds after it has been hit.

The function $$f$$ given by $$f(t) = 5 +30t - 32t^2$$ models the height of the ball in feet.

1. How high was the ball when it was hit? Where do you see this in the equation?
2. Suppose a second ball follows the same trajectory but is hit from 7 feet off the ground. Sketch the graph of the height of the second ball on the same axes.
3. Write an equation for a function $$g$$ that defines the height $$g(t)$$, in feet, of the second ball hit from 7 feet off the ground in terms of $$f(t)$$.

### Problem 5

1. Describe a horizontal translation of the line to a line that contains the two labeled points.
2. Describe a vertical translation of the line to a line that contains the two labeled points.
(From Unit 5, Lesson 1.)

### Problem 6

Does the function $$f$$ or the function $$g$$ fit the data better? Explain your reasoning.

(From Unit 5, Lesson 1.)