7.5: Graphing Cubed Root Functions
The next day, Mrs. Garcia assigns her student the cube root function \begin{align*}y = \sqrt[3]{(x + 1)}\end{align*}
Alendro says that because of the negative sign, all y values are negative. Therefore his graph is only in the third and fourth quadrants quadrant.
Dako says that his graph is in the third and fourth quadrants as well but it is also in the second quadrant.
Marisha says they are both wrong and that her graph of the function is in all four quadrants.
Which one of them is correct?
Guidance
A cubed root function is different from that of a square root. Their general forms look very similar, \begin{align*}y=a \sqrt[3]{xh}+k\end{align*}
x  y 

27  3 
8  2 
1  1 
0  0 
1  1 
8  2 
27  3 
For \begin{align*}y= \sqrt[3]{x}\end{align*}
Example A
Describe how to obtain the graph of \begin{align*}y= \sqrt[3]{x}+5\end{align*}
Solution: From the previous concept, we know that the +5 indicates a vertical shift of 5 units up. Therefore, this graph will look exactly the same as the parent graph, shifted up five units.
Example B
Graph \begin{align*}f(x)= \sqrt[3]{x+2}3\end{align*}
Solution: From the previous example, we know that from the parent graph, this function is going to shift to the left two units and down three units. The negative sign will result in a reflection.
Alternate Method: If you want to use a table (like in the previous concept), that will also work. Here is a table, then plot the points. \begin{align*}(h, k)\end{align*}
x  y 

6  5 
1  4 
2  3 
3  2 
10  1 
Example C
Graph \begin{align*}f(x)= \frac{1}{2} \sqrt[3]{x4}\end{align*}
Solution: The 4 tells us that, from the parent graph, the function will shift to the right four units. The \begin{align*}\frac{1}{2}\end{align*}
Using the graphing calculator: If you wanted to graph this function using the TI83 or 84, press \begin{align*}Y=\end{align*}
Important Note: The domain and range of all cubed root functions are both all real numbers.
Intro Problem Revisit If you graph the function \begin{align*}y = \sqrt[3]{(x + 1)}\end{align*}
Guided Practice
1. Evaluate \begin{align*}y= \sqrt[3]{x+4}11\end{align*}
2. Describe how to obtain the graph of \begin{align*}y= \sqrt[3]{x+4}11\end{align*}
Graph the following cubed root functions. Check your graphs on the graphing calculator.
3. \begin{align*}y= \sqrt[3]{x2}4\end{align*}
4. \begin{align*}f(x)=3 \sqrt{x}1\end{align*}
Answers
1. Plug in \begin{align*}x=12\end{align*}
\begin{align*}y= \sqrt[3]{12+4}11= \sqrt[3]{8}+4=2+4=2\end{align*}
2. Starting with \begin{align*}y= \sqrt[3]{x}\end{align*}
3. This function is a horizontal shift to the right two units and down four units.
4. This function is a reflection of \begin{align*}y= \sqrt[3]{x}\end{align*}
Explore More
Evaluate \begin{align*}f(x)=\sqrt[3]{2x1}\end{align*}

\begin{align*}f(14)\end{align*}
f(14) 
\begin{align*}f(62)\end{align*}
f(−62) 
\begin{align*}f(20)\end{align*}
f(20)
Graph the following cubed root functions. Use your calculator to check your answers.
 \begin{align*}y=\sqrt[3]{x}+4\end{align*}
 \begin{align*}y=\sqrt[3]{x3}\end{align*}
 \begin{align*}f(x)=\sqrt[3]{x+2}1\end{align*}
 \begin{align*}g(x)= \sqrt[3]{x}6\end{align*}
 \begin{align*}f(x)=2 \sqrt[3]{x+1}\end{align*}
 \begin{align*}h(x)=3 \sqrt[3]{x}+5\end{align*}
 \begin{align*}y=\frac{1}{2} \sqrt[3]{1x}\end{align*}
 \begin{align*}y=2 \sqrt[3]{x+4}3\end{align*}
 \begin{align*}y= \frac{1}{3} \sqrt[3]{x5}+2\end{align*}
 \begin{align*}g(x)=\sqrt[3]{6x}+7\end{align*}
 \begin{align*}f(x)=5 \sqrt[3]{x1}+3\end{align*}
 \begin{align*}y=4 \sqrt[3]{7x}8\end{align*}
Image Attributions
Description
Learning Objectives
Here you'll graph a cubed root function with and without a calculator.
Difficulty Level:
At GradeSubjects:
Concept Nodes:
Date Created:
Mar 12, 2013Last Modified:
Jun 04, 2015Vocabulary
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