Lesson 9

Every weekend, Elena takes a walk along the straight road in front of her house for 2 miles, then turns around and comes back home. Let’s assume Elena walks at a constant speed.

Here is a graph of the function $f$ that gives her distance $f(t)$, in miles, from home as a function of time $t$ if she walks 2 miles per hour.

1. Sketch a graph of the function $g$ that gives her distance $g(t)$, in miles, from home as a function of time $t$ if she walks 4 miles per hour.

2. Write an equation for $g$ in terms of $f$. Be prepared to explain why your equation makes sense.

Remember Clare on the Ferris wheel? In the table, we have the function $F$ which gives her height $F(t)$ above the ground, in feet, $t$ seconds after starting her descent from the top. Today Clare tried out two new Ferris wheels.

• The first wheel is twice the height of $F$ and rotates at the same speed. The function $g$ gives Clare's height $g(t)$, in feet, $t$ seconds after starting her descent from the top.
• The second wheel is the same height as $F$ but rotates at half the speed. The function $h$ gives Clare's height $h(t)$, in feet, $t$ seconds after starting her descent from the top.

$t$	$F(t)$	$g(t)$	$h(t)$
0	212
20	181
40	106
60	31
80	0

1. Complete the table for the function $g$.
2. Explain why there is not enough information to find the exact values for $h(20)$ and $h(60)$.
3. Complete as much of the table as you can for the function $h$, modeling Claire's height on the second Ferris wheel.
4. Express $g$ and $h$ in terms of $f$. Be prepared to explain your reasoning.

Function $k$ is a transformation of function $P$ due to a scale factor.

1. Write an equation for $k$ in terms of $P$.
2. On the same axes, graph the function $m$ where $m(x)=P(0.75x)$.
3. The highest point on the graph of $P$ is $(1,2)$. What is the highest point on the graph of a function $n$ where $n(x)=P(5x)$? Explain or show your reasoning.
4. The point furthest to the right on the graph of $P$ is $(4,0)$. If the point furthest to the right on the graph of a function $q$ is $(18,0)$, write a possible equation for $q$ in terms of $P$.

Here are two graphs showing the distance traveled by two trains $t$ hours into their journeys. What do you notice?

Where Train A traveled 25 miles in 1 hour, Train B traveled 25 miles in half the time. Similarly, Train A traveled 150 miles in 4 hours while Train B traveled 150 miles in only 2 hours. Train B is traveling twice the speed of Train A.

A train travelling twice the speed gets to any particular point along the track in half the time, so the graph for Train B is compressed horizontally by a factor of $\frac12$ when compared to the graph of Train A. If the function $f(t)$ represents the distance Train A travels in $t$ hours, then $f(2t)$ represents the distance Train B travels in $t$ hours, because Train B goes as far in $t$ hours as Train A goes in $2t$ hours.

If a different Train C were going one fourth the speed of Train A, then its motion would be represented by $s = f(0.25t)$ and the graph would be stretched horizontally by a factor of 4 since it would take four times as long to travel the same distance.