Describe the Transformation y=1/2 cube root of x
In the problem statement, you are asked to describe the transformation that occurs when you apply the function y = (1/2) * cube root of x to the input variable x. This involves explaining how the graph of the basic cube root function y = cube root of x is altered when it is scaled by a factor of 1/2 vertically. It involves discussing any changes to the shape, orientation, and position of the graph on the coordinate plane as a result of this scaling transformation.
$y = \frac{1}{2} \sqrt[3]{x}$
Answer
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.
