Describe the Transformation y=1/2 cube root of x

Solution:

Step 1:

Identify the basic function, which is $y=\sqrt[3]{x}$.

Step 2:

Combine the constant $\frac{1}{2}$ with the cube root to get $y=\frac{1}{2}\sqrt[3]{x}$.

Step 3:

Let $f(x)=\sqrt[3]{x}$ represent the parent function and $g(x)=\frac{1}{2}\sqrt[3]{x}$ represent the transformed function.

Step 4:

To understand the transformation, we look for the values of $a$, $h$, and $k$ in the general form $y=a\sqrt[3]{x-h}+k$.

Step 5:

Extract a factor of $1$ from the cube root to ensure the coefficient of $x$ inside the root is $1$: $y=\sqrt[3]{x}$.

Step 6:

Apply the same factor to the transformed function: $y=\frac{1}{2}\sqrt[3]{x}$.

Step 7:

Determine the values of $a$, $h$, and $k$ for the transformed function: $a=0.5$, $h=0$, $k=0$.

Step 8:

The horizontal shift is determined by the value of $h$. Since $h=0$, there is no horizontal shift.

Step 9:

The vertical shift is determined by the value of $k$. Since $k=0$, there is no vertical shift.

Step 10:

The sign of $a$ indicates if there is a reflection across the x-axis. Since $a$ is positive, there is no reflection.

Step 11:

The absolute value of $a$ indicates a vertical stretch or compression. Since $0<a<1$, there is a vertical compression.

Step 12:

To summarize the transformation, compare the parent function $f(x)$ and the transformed function $g(x)$: There is no horizontal or vertical shift, no reflection, and there is a vertical compression.

Step 13:

The transformation of the function is therefore a vertical compression by a factor of $\frac{1}{2}$.

Knowledge Notes:

The transformation of a function involves altering its shape, position, or orientation on a graph. The general form of a transformed function is $y=a\sqrt[3]{x-h}+k$, where:

• $a$ determines the vertical stretch or compression. If $|a|>1$, the graph is stretched; if $0<|a|<1$, the graph is compressed.

• $h$ determines the horizontal shift. If $h>0$, the graph shifts to the right; if $h<0$, the graph shifts to the left.

• $k$ determines the vertical shift. If $k>0$, the graph shifts up; if $k<0$, the graph shifts down.

• The sign of $a$ also indicates a reflection across the x-axis if it is negative.

In this problem, the transformation of the cube root function $y=\sqrt[3]{x}$ involves a vertical compression by a factor of $\frac{1}{2}$, which is represented by $g(x)=\frac{1}{2}\sqrt[3]{x}$. There are no horizontal shifts, vertical shifts, or reflections in this transformation.