12.1: Radical Transformations
This activity is intended to supplement Algebra I, Chapter 11, Lesson 1.
ID: 11574
Time Required: 15 minutes
Activity Overview
In this activity, students will use the transformational graphing application to examine how the square root function is transformed on the coordinate plane. As an extension, students will examine similar transformations on a cube root function.
Topic: Radical Functions
- Domain, Range
- Square Root
- Transformations
Teacher Preparation and Notes
- Students will need the Transformation Graphing application installed on their graphing calculators.
- To download the calculator application, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=11574 and select Transformation Graphing under the Applications header.
Associated Materials
- Student Worksheet: Radical Transformation, http://www.ck12.org/flexr/chapter/9621
- Transformation Graphing
Problem 1 – The General Radical Function
Students will examine the graph of \begin{align*}y=\sqrt{x}\end{align*} and determine the domain and range of the function.
Students will also conjecture if the graph is always in the first quadrant.
Problem 2 – Transformations
In Problem 2, students will change values in the general equation of a square root graph. Students will determine the domain and range of a square root function based on the general equation. They will revisit the question of where the graph lies in the plane. They will also describe the transformation performed on the graph by changing each variable.
Discussion Questions:
- How does each variable affect the graph of the function?
- How can we algebraically show the domain and range of the function?
- Why does the graph “stop” (no longer exist) on one side?
- How does the variable a affect the graph?
Extension – Cube Root Functions
In this problem, students will repeat the exploration above for the cube root function and determine domain, range, and transformations on the graph.
Note: The cube root can be entered by pressing MATH and selecting 5: \begin{align*}\sqrt[3]{\;\;}\end{align*}.
Solutions
- Domain: \begin{align*}x \ge 0\end{align*}; Range: \begin{align*}y \ge 0\end{align*}
- Sample Response: The function is not defined for values less than zero because the square root becomes negative.
- Sometimes (Incorrect answer is okay. This is a conjecture question.)
- Sample response: The graph is a straight line because the square root is multiplied by zero, making the function \begin{align*}y = 0\end{align*}.
- Sometimes (Now students should have the correct answer.)
- Sample responses must have \begin{align*}\sqrt{x-3}\end{align*}
- \begin{align*}x \ge -2\end{align*}
- \begin{align*}h\end{align*}
- Sample response must have \begin{align*}-2\end{align*} after square root
- \begin{align*}y \ge 3\end{align*}
- \begin{align*}k\end{align*}
- Sample response: Positive \begin{align*}4\end{align*} opens down (concave down); Negative \begin{align*}4\end{align*} opens up (concave up).
- Sample response: Positive \begin{align*}4\end{align*} is steeper than positive \begin{align*}2\end{align*}.
- Sample response: Flips the graph open up or open down. Makes the graph steeper as \begin{align*}|a|\end{align*} gets larger.
- \begin{align*}x \ge h\end{align*}
- \begin{align*}y \ge k\end{align*}
- Domain and range: All real numbers
- \begin{align*}h\end{align*} is a horizontal shift; \begin{align*}k\end{align*} is a vertical shift; \begin{align*}a\end{align*} makes the branches steeper and flips the graph.