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 gives the distance , in miles, of Tyler from his house minutes after 7:00 a.m.

1. Explain what means in this context.
2. On snowy days, Tyler’s school has a 2-hour delayed start time (120 minutes). The function gives Tyler’s distance , in miles, from home minutes after 7:00 a.m. with a 120-minute delayed start time. If , then what is the corresponding point on the function ?
3. Write an equation for in terms of .
4. A new function, , is defined as . Explain what this means in terms of Tyler’s distance from school.

Technology required. Here are the data for the population , in thousands, of a city decades after 1960 along with the graph of the function given by . Elena thinks that shifting the graph of up by 50 will match the data. Han thinks that shifting the graph of up by 60 and then right by 1 will match the data.

1. What functions define Elena's and Han's graphs?
2. Use graphing technology to graph Elena's and Han's proposed functions along with .
3. Which graph do you think fits the data better? Explain your reasoning.

Data and function on coordinate grid, origin “O”. Horizontal axis, time, decades after 1960, from 0 to 6 by 1's. Vertical axis, population, thousands,, from 0 to 150 by 25's. Function starts at 0 comma 25 and moves upwards and to the right, passing approximately through 1 comma 29, 2 comma 35, 3 comma 42, 4 comma 50, 5 comma 60 and 6 comma 71. Approximate data points at 0 comma 70, 1 comma 73, 2 comma 88, 3 comma 90, 4 comma 105, 5 comma 120 and 6 comma 130.  

Here is a graph of for a function .

Sketch the graph of .

Describe how to transform the graph of to the graph of :

Function f and g on x y coordinate plane. X axis from negative 6 to 8 by 2's. Y axis from negative 2 to 3 by 1's. Function f passes through negative 4 comma 3, curves downwards and then upwards at negative 1 comma 1 until 2 comma 3, curves downwards until 5 comma 1, then curves upwards until 8 comma 3. Function g passes through negative 6 comma 1, curves downwards until negative 3 comma negative 1, curves upwards until 0 comma 1, curves downwards until 3 comma negative 1, curves upwards until 6 comma 1.  

1. using only translations
2. using a reflection and a translation

Here is a graph of function and a graph of function . Express in terms of using function notation.

A graph of function f on a coordinate plane. X axis from negative 10 to 5, by 5’s. Y axis from negative 15 to 5, by 5’s. From left to right, the function begins in the second quadrant, moves downward and to the right, crossing the x axis at approximately negative 6, continues downward reaching minimum near negative 4 point 5 comma negative 7. It then curves up crossing the y axis at 1, curves back down until it reaches about 2 point 5 comma 0. It then curves back upward and to the right, crosses the x axis at 2 point 5, and continues to move upward and to the right, ending in the first quadrant.

A graph of function g on a coordinate plane. X axis from negative 10 to 5, by 5’s. Y axis from negative 15 to 5, by 5’s. From left to right, the function begins in the second quadrant, moves downward and to the right, crossing the x axis at approximately negative 6, continues downward reaching minimum near negative 4 point 5 comma negative 10. It then curves up crossing the y axis at negative 2, curves back down until it reaches 3 comma negative 3. It then curves back upward and to the right, crosses the x axis at 4, and continues to move upward and to the right, ending in the first quadrant.