# Lesson 3

More Movement

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

### Problem 1

Here is a graph of \(f\) and a graph of \(g\). Express \(g\) in terms of \(f\) using function notation.

### Problem 2

Tyler leaves his house at 7:00 a.m. to go to school. He walks for 20 minutes until he reaches his school, 1 mile from his house. The function \(d\) gives the distance \(d(t)\), in miles, of Tyler from his house \(t\) minutes after 7:00 a.m.

- Explain what \(d(5)=0.25\) means in this context.
- On snowy days, Tyler’s school has a 2 hour delayed start time (120 minutes). The function \(s\) gives Tyler’s distance \(s(t)\), in miles, from home \(t\) minutes after 7:00 a.m. with a 120 minute delayed start time. If \(d(5)=0.25\), then what is the corresponding point on the function \(s\)?
- Write an expression for \(s\) in terms of \(d\).
- A new function, \(n\), is defined as \(n(t)=d(t + 60)\) explain what this means in terms of Tyler’s distance from school.

### Problem 3

*Technology required.* Here are the data for the population \(f\), in thousands, of a city \(d\) decades after 1960 along with the graph of the function given by \(f(d) = 25 \boldcdot (1.19)^d\). Elena thinks that shifting the graph of \(f\) up by 50 will match the data. Han thinks that shifting the graph of \(f\) up by 60 and then right by 1 will match the data.

- What functions define Elena's and Han's graphs?
- Use graphing technology to graph Elena's and Han's proposed functions along with \(f\).
- Which graph do you think fits the data better? Explain your reasoning.

### Problem 4

Here is a graph of \(y = f(x+2)-1\) for a function \(f\).

Sketch the graph of \(y = f(x)\).

### Problem 5

Describe how to transform the graph of \(f\) to the graph of \(g\):

- using only translations
- using a reflection and a translation

### Problem 6

Here is a graph of function \(f\) and a graph of function \(g\). Express \(g\) in terms of \(f\) using function notation.