Lesson 11

Making a Model for Data

Let’s model with functions.

Problem 1

A line \(\ell\) is defined by the equation \(f(x) = 2x - 3\).

  1. Line \(m\) is the same as line \(l\), but shifted 1 unit right. What is an equation for a function \(g\) that defines the line \(m\)?
  2. Line \(n\) is the same as line \(m\), but shifted 2 units up. What is an equation for a function \(h\) that defines the line \(n\)?
  3. What is the relationship between \(f\) and \(h\)?
(From Unit 5, Lesson 2.)

Problem 2

The functions \(g\) and \(f\) are related by the equation \(g(x) = f(\text-x) + 3\). Which sequence of transformations will take the graph of \(f\) to the graph of \(g\)?

(From Unit 5, Lesson 4.)

Problem 3

The function \(f\) is linear. Can \(f\) be an odd function? Explain how you know

(From Unit 5, Lesson 5.)

Problem 4

Technology required. The function \(f\) is given by \(f(x) = x^3 + 1\). Kiran says that \(f\) is odd because \((\text-x)^3 = \text-x^3\)

  1. Do you agree with Kiran? Explain your reasoning.
  2. Graph \(f\), and use the graph to decide whether or not \(f\) is an odd function. 
(From Unit 5, Lesson 6.)

Problem 5

Here are graphs of three functions \(f\), \(g\), and \(h\) given by \(f(x) = (x-1)^2\), \(g(x) = 2(x-1)^2\) and \(h(x) = 3(x-1)^2\).

Graph of 3 functions.

Identify which function matches each graph. Explain how you know.

(From Unit 5, Lesson 8.)

Problem 6

Technology required. Describe how to transform the graph of \(f(x) = x^2\) into the graph of \(g(x) = 4(3x-1)^2 + 5\). Check your response by graphing \(f\) and \(g\).

(From Unit 5, Lesson 9.)

Problem 7

Let \(p\) be the price of a T-shirt, in dollars. A company expects to sell \(f(p)\) T-shirts a day where \(f(p) = 50 - 4p\). Write a function \(r\) giving the total revenue received in a day.

(From Unit 5, Lesson 10.)

Problem 8

A population of 80 single-celled organisms is tripling every hour. The population as a function of hours since it is measured, \(h\), can be represented by \(g(h) =80 \boldcdot 3^h\).

Which equation represents the population 10 minutes after it is measured?


\(g(10) =80 \boldcdot 3^{10}\)


\(g(0.1) =80 \boldcdot 3^{0.1}\)


\(g(\frac16) =80 \boldcdot 3^\frac16\)


\(g(6) =80 \boldcdot 3^6\)

(From Unit 4, Lesson 3.)