Unit 7 Variation and Square Root Functions

Lesson 1

Learning Focus

Make observations about the domain, range, and rate of change of the square root function.

Lesson Summary

In this lesson, we extended our understanding of square roots by defining the square root function, $f (x) = \sqrt{x}$ , and exploring the key features of its graph.

Lesson 2

Learning Focus

Examine how changes in the quantities of a context transform the graph of the square root function that models the context.

Lesson Summary

In this lesson, we learned the graph of a square root function can be transformed in the same way as linear, exponential and quadratic functions are transformed, including horizontal and vertical translations and dilations. We also reviewed conditions that could be used to verify if two quantities are proportional to each other. In this lesson, we called these relationships direct variations and found that if we define the quantities carefully, we can find direct variations between such quantities as the “square root of the length of a pendulum” and “the period of a pendulum.”

Lesson 3

Learning Focus

Solve equations and systems of equations that involve square root expressions.

Lesson Summary

In this lesson, we learned how to solve equations that include square root expressions. The process we used often required us to recall how to solve quadratic equations. Sometimes the process introduced extraneous solutions, so it is important to check the solutions to make sure they satisfy the original equation.

Lesson 4

Learning Focus

Analyze the domain and range of transformed square root functions to assist in sketching their graphs.

Given a table or graph, write the equation of the square root function that fits it.

Reexamine assumptions and strategies for solving square root equations.

Lesson Summary

In today’s lesson, we learned how to find the domain and range of a transformed square root function and how to use that information to sketch a graph of the function. In addition, we learned how to fit a square root function to a table of data or to the points on a graph. We also examined our strategies for solving equations that include square roots and observed that a square root equation can have more than one solution.

Lesson 5

Learning Focus

Identify and create inverse variation functions, tables, and graphs to model real-world situations.

Lesson Summary

In this lesson, we learned about inverse variations, functions in which the quantities are said to vary inversely because doubling one quantity cuts the other in half, tripling the quantity cuts the other quantity in thirds, etc.

Lesson 6

Learning Focus

Analyze contexts to identify the key features of an inverse variation.

Lesson Summary

In this lesson, we continued to explore inverse variation functions in geometric contexts and found that we may need to strategically choose the quantities related by a function in order to reveal a potential inverse variation relationship. We also learned to be careful when examining the shape of a graph as potentially displaying an inverse function relationship.

Lesson 7

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.

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.