# Lesson 9

Scaling the Inputs

• Let’s use scale factors in different ways.

### 9.1: Out and Back

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.

### 9.2: A New Set of Wheels

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.

### 9.3: The Many Transformations of a Function $P$

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 $$P$$.

What transformation takes $$f(x)=2x(x-4)$$ to $$g(x)=8x(x-2)$$?

### Summary

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.