Lesson 7 More Variation Practice Understanding

Learning Focus

Examine properties of graphs of the form $y = \frac{k}{x}$ over the domain of all real numbers for which the function is defined.

Solve systems of equations involving two square root and/or inverse variation equations using an appropriate method.

What are the properties of an inverse variation function that is defined for all real numbers (except $0$ )?

When the solution to a system of equations is not an ordered-pair of integers, how can I use graphs and successive approximations in a table to find a reasonable solution to the system?

Open Up the Math: Launch, Explore, Discuss

We have found that equations of the form $y = \frac{k}{x}$ are inverse variations, since the product $x y$ always equals a constant, $k$ . Since we have been exploring inverse variations in context, the domain of the function has been restricted to positive integers for discrete contexts or positive real numbers for continuous contexts. What would happen if we didn’t restrict the value of $x$ to positive numbers?

1.

Sketch a graph of $y = \frac{k}{x}$ with the least restrictive domain possible. For simplicity, you can choose $k = 1$ to get one member of this family of functions.

2.

What would the domain and range of this function be?

3.

What are some features you would want to include in a description of a general inverse variation function graph?

4.

How do different values of $k$ affect the shape of the graph?

5.

What if $y$ varied inversely with respect to the quantity $x + 5$ ? Sketch a graph of this function, that is, $y = \frac{k}{x + 5}$ . For simplicity, you can choose $k = 1$ to get one member of this family of functions.

6.

Find the points of intersection for the two graphs defined by this system: ${\begin{matrix} y = \frac{1}{x} \\ y = \frac{3}{x + 5} \end{matrix}$

You may use a graphical, algebraic, or successive approximation (guess and check) approach to find the solution to this system. What convinces you that you have found all solutions?

7.

Find the points of intersection for the two graphs defined by this system: ${\begin{matrix} y = \frac{2}{x} \\ y = \frac{1}{2} \sqrt{x} \end{matrix}$

You may use a graphical, algebraic, or successive approximation (guess and check) approach to find the solution to this system. What convinces you that you have found all solutions?

8.

Find the points of intersection for the two functions defined by the table and graph given.

You may use a graphical, algebraic, or successive approximation (guess and check) approach to find the solution to this system. What convinces you that you have found all solutions?

$x$	$y$
$0$	$0$
$1$	$2$
$4$	$4$
$9$	$6$
$16$	$8$
$25$	$10$
$36$	$12$

Pause and Reflect

Direct and inverse variations describe many real-world relationships between two quantities. Ancient civilizations assumed that the circumference of a circle varied directly with its diameter—that is, if you doubled the diameter you would double the circumference, if you tripled the diameter you would triple the circumference, etc. Therefore, $C = k d$ . The constant of proportionality, $k$ , was found experimentally to be approximately $3$ , or exactly $π = 3.14159 . . .$

Scientists during the Renaissance assumed that the pressure of a gas, such as steam, would vary directly with temperature, $P = k T$ , and vary inversely with the volume in which the gas was contained, $P = \frac{k}{V}$ . This led to the discovery of the Gas Laws, the invention of the Kelvin temperature scale in which these proportionality statements would hold true, and the recognition of absolute $0$ , or the temperature at which the molecules of a substance would not be in motion.

9.

An empty plastic water bottle with its lid replaced contains a constant amount of air. If left in a car, the bottle will make popping noises as it expands or contracts as the temperature inside the car changes during the day. The popping and crackling are due to the changing pressure of the air (a gas) on the walls of the bottle containing the air. This is an example of the Gas Laws.

a.

Explain the meaning of the gas law $P = k T$ in this context.

b.

Explain the meaning of the gas law $P = \frac{k}{V}$ in this context.

10.

The area of a circle varies directly with the square of its radius. The constant of proportionality is $π$ . Give a symbolic rule for this statement.

11.

The volume of a sphere varies directly with the cube of its radius. The constant of proportionality is $\frac{4}{3} π$ . Give a symbolic rule for this statement.

12.

The period of a pendulum varies directly with the square root of its length. Give a symbolic rule for this statement.

13.

The intensity of light varies inversely with the square of the distance from the the light source. Give a symbolic rule for this statement.

Ready for More?

Do some research on how scientific laws, such as the Gas Laws, the laws of gravitation, Newton’s Law of Cooling, light intensity, etc., emerged from the assumption that two quantities either vary directly or inversely with each other. Why do these laws make intuitive sense when you relate them to ideas about direct and inverse variation?

Takeaways

The key features of the graph of an inverse variation function of the form $y = \frac{k}{x}$ with unrestricted domain include:

Strategies for solving systems of equations that involve non-linear functions include:

Adding Notation, Vocabulary, and Conventions

Horizontal asymptotes

Vertical asymptotes

Vocabulary

asymptote
horizontal asymptote
vertical asymptote
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we examined the key features of functions of the form $y = \frac{k}{x}$ and found that the graph contains vertical and horizontal asymptotes that determine the end-behavior of the graph, as well as the behavior of the graph near $x = 0$ . We also solved systems of equations that involved non-linear functions, such as square root functions and/or inverse variations. The solutions to the systems can be approximated graphically. The approximate solutions can be improved by successive approximations in a table. The exact solutions can often be found algebraically.

Retrieval

1.

In an experiment, an event is the outcome that we are interested in occurring.

The probability of an event $A$ , written as $P (A)$ , is defined as

$P (A) = \frac{Number of favorable outcomes}{Total number of possible outcomes}$

Nikki loves red candy. There are $6$ yellow, $10$ green, $9$ orange, $7$ blue, and $4$ red candies in a brown paper sack. Without looking, Nikki selected a candy out of the sack. What is the probability that she selected a red one?

2.

Identify the type of function in each table. Justify your answer.

Choices for types of functions: linear but not direct variation, linear and direct variation, inverse variation, quadratic, and exponential.

a.

$x$	$y$
$- 4$	$- 9$
$- 3$	$- 12$
$- 2$	$- 18$
$2$	$18$
$3$	$12$
$4$	$9$

b.

$x$	$y$
$- 3$	$0$
$- 2$	$- 5$
$- 1$	$- 8$
$0$	$- 9$
$1$	$- 8$
$2$	$- 5$