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.

1. Explain what \(d(5)=0.25\) means in this context.
2. 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 \(s\)?
3. Write an equation for \(s\) in terms of \(d\).
4. A new function, \(n\), is defined as \(n(t)=d(t + 60)\). Explain what this means in terms of Tyler’s distance from school.

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.

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 \(f\).
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 \(y = f(x+2)-1\) for a function \(f\).

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

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

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 \(f\) and a graph of function \(g\). Express \(g\) in terms of \(f\) 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.