# 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.

- 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.
- Explain what \(H(6.5) = 65\) means in this context. Why is this a reasonable value for the function?
- 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 \(G\).
- 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.

- Explain what \(P(30)=350\) means in this context.
- 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 \(B\)?
- Write an expression for \(B\) in terms of \(P\).

### Problem 3

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

- On the same axes, sketch a graph of \(g(x) = f(x) + 2\).
- On the same axes, sketch a graph of \(h(x) = f(x+2)\).
- 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.

- How high was the ball when it was hit? Where do you see this in the equation?
- 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.
- 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

- Describe a horizontal translation of the line to a line that contains the two labeled points.
- Describe a vertical translation of the line to a line that contains the two labeled points.

### Problem 6

Does the function \(f\) or the function \(g\) fit the data better? Explain your reasoning.