Lesson 2 Root Variation Solidify Understanding

Learning Focus

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

How can we model the motion of swinging on a swing?

What quantities might affect the length of time it takes to swing forward and back, returning to the starting position?

Open Up the Math: Launch, Explore, Discuss

Tehani, Taska, and Trevor have entered an engineering competition in which students have to redesign an Earth-based amusement park full of rides so the rides will feel and behave the same when installed on other planets, specifically Mars or Jupiter. Because the pull of gravity on each of these three planets is different, there are a lot of mathematical challenges involved in recreating the amusement park rides. Tehani’s team has been assigned to redesign the swings that will appear throughout the park, some smaller swings in the children’s portion of the park, a large swing in the adult section of the park, and some swings that replicate swinging vines in the Tarzan: Escape from the Jungle ride.

Trevor has found that that the period, $T$ (one forward and back motion of a swing), is related to the length, $L$ , of the swing by the formula $T = k \sqrt{L}$ . In this formula, the constant $k$ depends on the acceleration of gravity acting on the weight of the object carried by the swing, and therefore the value of $k$ will change from planet to planet. On Earth, $k = 1.2$ when the length of the swing is measured in feet and the period is measured in seconds.

1.

Calculate the period of a swing that is $9 feet$ long.

2.

If you double the length of the swing, will you double the length of time it takes for one period?

Why or why not (that is, provide evidence for your claim)?

3.

How much longer do you need to make the $9 -foot$ long swing to double its period?

4.

How many times longer do you need to make the $9 -foot$ long swing to double its period?

5.

How many times longer do you need to make the $9 -feet$ long swing to triple its period?

6.

In general, if you want to extend the period of the $9 -foot$ long swing by a factor of $n$ , how much longer do you need to make the swing?

Taska is working on the redesign of the swings for the park on Jupiter. The formula for the period of a swing on Jupiter is $T = 0.6 \sqrt{L}$ .

7.

Describe how a ride on a swing transported from Earth to Jupiter without adjustment would compare to the ride experienced on the same swing on Earth, where the period is given by $T = 1.2 \sqrt{L}$ .

8.

Trevor has found that included in their instructions they were given a graph of the period of a swing as a function of its length on the planet Earth. That is, this is a graph of $T = 1.2 \sqrt{L}$ . Draw the graph of the period of a swing for the planet Jupiter on this same coordinate grid. Remember that the formula for the period of a swing on Jupiter is $T = 0.6 \sqrt{L}$ .

9.

Add the graph of the period of a swing for Mars to this graph of the period of a swing on the Earth, given that the formula for the period of a swing on Mars relative to its length is $T = 2.4 \sqrt{L}$ .

10.

Tehani has inspected Taska’s work and has several changes she wants made to various swings around the playground. Since each swing that needs to be adjusted is of a different length, she has written her instructions using $x$ to represent the original length of the swing before the desired adjustment is made. Interpret the following notation in terms of what Taska is being asked to adjust. Be specific, in terms of the context, by describing what quantity needs to be changed and by how much, including units.

a.

$T = 0.6 \sqrt{x + 5}$

b.

$T = 0.6 \sqrt{x} + 2$

c.

$T = 0.6 \sqrt{2 x}$

d.

$T = 3 \cdot 0.6 \sqrt{x}$

11.

The graph and a table for the period $T = 0.6 \sqrt{x}$ of a swing of length $x$ on Jupiter is given below. Complete either the table or the graph for each of the following related functions. Use descriptions of the contextual changes to the swing (see problem 10) to justify the coordinates you list in the table or the shape and location of the new graphs.

$T = 0.6 \sqrt{x + 5}$
$T = 0.6 \sqrt{x} + 2$
$T = 0.6 \sqrt{2 x}$
$T = 3 \cdot 0.6 \sqrt{x}$

$x$	$T = 0 . 6 \sqrt{x}$	$T = 0 . 6 \sqrt{x + 5}$	$T = 0 . 6 \sqrt{x} + 2$	$T = 0 . 6 \sqrt{2 x}$	$T = 3 \cdot 0 . 6 \sqrt{x}$
$0$	$0$
$5$	$1.34$
$10$	$1.90$
$15$	$2.32$
$20$	$2.68$

The Language of Proportionality Relationships (or Variation)

You studied proportionality relationships in Grade 7. One way to tell if two quantities are related by a proportionality relationship is to see how a change in the input affects the change in the output. In general, if doubling the input doubles the output, tripling the input triples the output, cutting the input in half cuts the output in half, etc., then the input and output quantities are related by a direct proportionality relationship. Mathematicians use language, such as, “The output quantity is proportional to the input quantity,” or, “the output quantity varies directly with the input quantity” to describe such relationships.

12.

If you are running at a rate of $4 miles per hour$ , your distance as a function of time is given by $d = 4 t$ . Is this relationship a direct variation relationship?

Why or why not?

13.

Is the period of a swing relationship $T = k \sqrt{L}$ , a direct variation relationship?

Why or why not?

While the period of a swing equation is not a direct variation, scientists use the language of proportionality to describe this type of relationship using the language, “The period of a swing is proportional to the square root of the length.” You might wonder why this is called a proportionality relationship since doubling the length of the swing does not double the period, etc. However, this statement is not about the relationship between the quantities period and length, but rather the quantities period and square root of the length.

Table 1:

Length	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$	$18$	$20$	$22$	$24$
Period	$1.697$	$2.4$	$2.939$	$3.394$	$3.794$	$4.157$	$4.490$	$4.8$	$5.091$	$5.367$	$5.628$	$5.879$

Graph 1:

Table 2:

$\sqrt{Length}$	$Period$
$1.414$	$1.697$
$2$	$2.4$
$2.45$	$2.939$
$2.828$	$3.394$
$3.162$	$3.794$
$3.464$	$4.157$
$3.742$	$4.490$
$4$	$4.8$
$4.243$	$5.091$
$4.472$	$5.367$
$4.690$	$5.628$
$4.90$	$5.878$

Graph 2:

14.

Examine the two tables and graphs given. Which table illustrates a direct variation?

How do you know?

15.

Which graph illustrates a direct variation?

How do you know?

Ready for More?

Based on the information in the last part of the task, is there a way that you can state and justify a direct variation relationship for the equation for the area of a circle: $A = π r^{2}$ ?

That is, varies directly with .

Use a table or a graph to justify your direct variation statement.

Takeaways

The square root function $f (x) = \sqrt{x}$ can be transformed in the same way as other functions we have seen.

For example:

$f (x) = \sqrt{x} + k$ .
$f (x) = \sqrt{x + k}$ .
$f (x) = k \sqrt{x}$ .
$f (x) = \sqrt{k x}$ .

The following tests can be used to determine if two quantities $x$ and $y$ are proportional to each other (or in other words, are in a direct variation relationship):

An equation like $y = k \sqrt{x}$ is not a direct variation in the quantities $x$ and $y$ , but is a direct variation

Vocabulary

constant of proportionality/ constant of variation
direct variation
Bold terms are new in this lesson.

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.”

Retrieval

1.

An astronomer, Dr. Astral, wrote a system of two equations to see if he could predict when the paths of the objects he was studying would intersect. He solved the system by setting the equations equal to each other and solving for $x$ . He then used substitution to find $y$ . After solving the systems, he looked at the graphs of the equations to check if he was right.

Set the equations equal to each other and solve for $x$ .

$[\begin{matrix} y = x^{2} - 8 x + 12 \\ y = - 2 x + 7 \end{matrix}$

Check your answers by graphing.

2.

Find the product. ${(3 x - 5)}^{2}$