Lesson 4 Root Knowledge Practice Understanding

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.

How can you predict the domain, range, and shape of a transformed square root function without examining the graph?

Open Up the Math: Launch, Explore, Discuss

PART 1: DOMAIN OF SQUARE ROOT FUNCTIONS

In the previous task, Trevor and Mateo had to think about the domain of a square root function in order to use its graph to solve a system of equations. Use algebraic reasoning to determine the domain and range of each of the following square root functions, then sketch its graph on the coordinate grid. Check your results by examining a graph produced using graphing technology.

1.

$f (x) = \sqrt{x - 4}$

domain:

range:

2.

$g (x) = \sqrt{4 - x}$

domain:

range:

3.

$h (x) = \sqrt{4 x} + 4$

domain:

range:

4.

$k (x) = 4 \sqrt{4 x - 4} + 4$

domain:

range:

PART 2: ADDITIONAL PRACTICE WITH SOLVING SQUARE ROOT EQUATIONS

As you noticed in the previous task, you need to carefully consider the order to undo the operations in an equation during the process of solving it.

5.

List the order you would undo the operations in this equation: $\sqrt{5 x + 6} - 2 = x$ .

6.

Solve this equation following the strategy listed in problem 5. Be sure to check for extraneous solutions.

$\sqrt{5 x + 6} - 2 = x$

7.

Solve the following system of equations: ${\begin{matrix} y = \sqrt{2 x - 1} \\ y = 2 + \sqrt{x - 4} \end{matrix}$

Pause and Reflect

PART 3: WRITING EQUATIONS OF SQUARE ROOT FUNCTIONS GIVEN A TABLE OR A GRAPH

Write an explicit rule for each of the following functions defined by either a table or a graph.

8.

$x$	$y$
$- 1$	$0$
$0$	$1$
$3$	$2$
$8$	$3$
$15$	$4$
$24$	$5$
$35$	$6$
$48$	$7$

9.

$x$	$y$
$0$	$0$
$2$	$1$
$8$	$2$
$18$	$3$
$32$	$4$
$50$	$5$
$72$	$6$
$98$	$7$

10.

Ready for More?

Write the equation of a transformed square root function that involves two or more transformations. Then create a table of data for their functions, using convenient values for $x$ so that all points in the data table have integer coordinates. Exchange your table with your partner and see if they can figure out the function rule that generated your data and if you can figure out the function rule that generated their data.

Takeaways

To find the domain of a square root function:

To find the endpoint of the square root function:

To find the range of the square root function:

To graph the square root function:

To avoid erroneous assumptions about the solutions to an equation that involves a square root expression, it should be observed that:

To fit a square root function to a table or a graph:

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.

Retrieval

1.

Jasper and his family are spending spring break at Myrtle Beach. They live $420 miles$ away. When Jasper and his father checked out the vacation rental earlier in the month, the trip took $6 hours$ , and Jasper figured out that their average speed was $70 mph$ . This time, the drive from home to the beach seemed like it took forever because Jasper’s younger siblings needed to stop often. It took $8.5 hours$ from the time Jasper’s family left their home until they parked in front of the rental cottage.

a.

Calculate the average speed of the second trip in miles per hour.

b.

On the second trip, the distance remained the same but the trip took longer. If distance remains the same but time increases, what happens to the average speed?

c.

What kind of relationship is this?

2.

Find the length of the pre-image and the image. Name the type of transformation. Indicate whether the transformation preserved distance or did not preserve distance.

Segment $A B$ defined by $A (- 9, - 15) B (- 21, - 10)$

a.

Segment $A^{'} B^{'}$ where $A^{'} (x, y) \to (x + 4, y - 2)$ and $B^{'} (x, y) \to (x + 4, y - 2)$

b.

Segment $A^{″} B^{″}$ where $A^{″} (x, y) \to (10 x, 10 y)$ and $B^{″} (x, y) \to (10 x, 10 y)$