# Lesson 9

Scaling the Inputs

### Problem 1

Here are graphs of functions $$f$$ and $$g$$. For each, determine the value of $$k$$ so that $$g(x)=f(kx)$$.

### Solution

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### Problem 2

Let $$f(x) = x(x-5)(x+2)(x+5)$$. Decide if the reasoning about each of the following functions is correct. Explain your reasoning.

1. Andre says that $$g(x) = 0.1x(0.1x-5)(0.1x+2)(0.1x+5)$$ is obtained from $$f$$ by scaling the inputs by a factor of 0.1.
2. Clare says this graph is a vertical shift of the graph of $$f$$ down 100 units.

3. Diego says the graph of $$k(x) = \text-x(x-5)(x+2)(x+5)$$ is the reflection of the graph of $$f$$ over the $$y$$-axis.

### Solution

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### Problem 3

A bacteria population, in thousands, is modeled by the function $$f(d) = 30 \boldcdot 2^d$$ where $$d$$ is the number of days since it was first measured. The function $$g$$ gives the bacteria population, in thousands, $$w$$ weeks after it was first measured. Express $$g$$ in terms of $$f$$. Explain your reasoning.

### Solution

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### Problem 4

The height of a hot air balloon, in feet, $$m$$ minutes after takeoff is modeled by the function $$f(m) = 16m$$.

1. How many minutes does it take for the balloon to reach 200 feet?
2. Another balloon takes off 5 minutes later and rises at the same speed. Write an equation for the function $$g$$, where $$g(t)$$ is the height, in feet, of this balloon in terms of $$m$$. Explain your reasoning.
3. Sketch graphs of the two functions $$f$$ and $$g$$.

### Solution

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(From Unit 5, Lesson 3.)

### Problem 5

Here is the graph of a function $$f$$.

Reflecting $$f$$ across the $$x$$-axis and then across the vertical line $$y=1$$ takes the graph of $$f$$ back to itself. Tyler says that this means $$f$$ is an odd function. Do you agree with Tyler? Explain your reasoning.

### Solution

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(From Unit 5, Lesson 5.)

### Problem 6

The population of sloths in an area has been increasing by 5% each year since 2000. Let $$P$$ model the population $$P(t)$$, in thousands, of sloths $$t$$ years after the year 2000. The graph of $$p(t) = 1.05^t$$ has a general shape that fits the data. Find a scale factor $$k$$ so that $$P(t) = kp(t)$$ fits the data.

years (since 2000) population (in thousands)
5 15.7
8 18.2
10 20.0
12 22.1
15 25.6
19 33.1

### Solution

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(From Unit 5, Lesson 8.)