# 12.1: Radical Transformations

**At Grade**Created by: CK-12

*This activity is intended to supplement Algebra I, Chapter 11, Lesson 1.*

## Problem 1 – The General Radical Function

Graph the equation \begin{align*}y = \sqrt{x}\end{align*}**TRACE** key to observe the coordinate values for points on the graph.

- What is the domain and range of the function?
- Why does the graph “stop” at the origin?
- When is the following statement true?

The graph of the square root function is completely in the first quadrant.

\begin{align*}\text{Sometimes} && \text{Always} && \text{Never}\end{align*}

## Problem 2 – Transformations

Start the **Transformation Graphing** application by pressing **APPS** and selecting **Transfrm**.

Now, press \begin{align*}Y=\end{align*} and enter \begin{align*}A \surd (X-B)+C\end{align*} into \begin{align*}Y_1\end{align*}.

Press **ZOOM** and select **6:ZStandard**. Notice the displayed equation. The values of \begin{align*}A, B\end{align*} and \begin{align*}C\end{align*} may be changed by using the arrow and number keys.

4. What does the graph look like when all three variables equal zero? Why?

5. Based on your exploration, when is the following statement true?

The graph of the square root function is completely in the first quadrant.

\begin{align*}\text{Sometimes} && \text{Always} && \text{Never}\end{align*}

Continue to manipulate the values of \begin{align*}A, B\end{align*} and \begin{align*}C\end{align*} on the calculator to help answer Questions 6-16

6. Find two functions whose domain is \begin{align*}x \ge 3\end{align*}.

7. What is the domain of the function \begin{align*}f(x) =4 \sqrt{x+2} - 3\end{align*} ? Check using the graph.

8. Changing which variable will create a horizontal shift?

9. Find two functions whose range is \begin{align*}y \ge -2\end{align*}.

10. What is range of the function \begin{align*}f(x) =4 \sqrt{x+2}-3\end{align*} ? Check using the graph.

11. Changing which variable will create a vertical shift?

12. What is the difference between the graphs of \begin{align*}f(x) =4 \sqrt{x+2}-3\end{align*} and \begin{align*}g(x) = -4 \sqrt{x+2}-3\end{align*} ?

13. What is the difference between the graphs of \begin{align*}f(x) =4 \sqrt{x+2}-3\end{align*} and \begin{align*}g(x) = 2 \sqrt{x+2}-3\end{align*} ?

14. What effect does the variable \begin{align*}a\end{align*} have on the graph?

15. What is the domain of the function using the general equation \begin{align*}y = \sqrt{x-h}+k\end{align*} ?

16. What is the range of the function using the general equation \begin{align*}y = \sqrt{x-h}+k\end{align*} ?

## Extension – Cube Root Functions

Press \begin{align*}Y=\end{align*} and enter \begin{align*}A \sqrt[3]{ \ }(x-B)+C\end{align*} into \begin{align*}Y_1\end{align*}.

Change the values of the variables \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*}, and observe the effects of the changes on the graph.

17. What is the domain and range of the function in terms of the general equation?

18. Describe the effects of changing each variable on the graph.

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