To understand how to graph the function [tex]\( y = \sqrt[3]{2x} \)[/tex], let's go through a step-by-step process to plot this function manually.
### Step 1: Understand the Function
The given function is [tex]\( y = \sqrt[3]{2x} \)[/tex], which can also be written as [tex]\( y = (2x)^{1/3} \)[/tex]. This is a cubic root function. Key properties of the cubic root function include:
- It is defined for all real numbers [tex]\( x \)[/tex].
- The cubic root function is odd, which means [tex]\( \sqrt[3]{-a} = -\sqrt[3]{a} \)[/tex].
- It passes through the origin (0, 0), since [tex]\( \sqrt[3]{0} = 0 \)[/tex].
### Step 2: Identify Key Points
Select a few key points to plot that will help shape the graph:
- When [tex]\( x = 0 \)[/tex]:
[tex]\( y = \sqrt[3]{2 \cdot 0} = \sqrt[3]{0} = 0 \)[/tex]
- Point: (0, 0)
- When [tex]\( x = 1 \)[/tex]:
[tex]\( y = \sqrt[3]{2 \cdot 1} = \sqrt[3]{2} \approx 1.26 \)[/tex]
- Point: (1, 1.26)
- When [tex]\( x = -1 \)[/tex]:
[tex]\( y = \sqrt[3]{2 \cdot (-1)} = \sqrt[3]{-2} \approx -1.26 \)[/tex]
- Point: (-1, -1.26)
- When [tex]\( x = 8 \)[/tex]:
[tex]\( y = \sqrt[3]{2 \cdot 8} = \sqrt[3]{16} \approx 2.52 \)[/tex]
- Point: (8, 2.52)
- When [tex]\( x = -8 \)[/tex]:
[tex]\( y = \sqrt[3]{2 \cdot (-8)} = \sqrt[3]{-16} \approx -2.52 \)[/tex]
- Point: (-8, -2.52)
You can choose more points for a more detailed graph, especially near 0 and in positive and negative regions.
### Step 3: Sketch the Graph
- Plot the key points on a coordinate plane.
- Since the function is continuous and smooth, connect these points with a smooth curve.
- Because we know the cubic root function is symmetric with respect to the origin (odd function), ensure the graph mirrors on the negative side of the x-axis.
### Graph Characteristics
- The graph will start from the left end extending through the origin and to the right.
- For positive [tex]\( x \)[/tex] values, [tex]\( y \)[/tex] increases but more slowly as [tex]\( x \)[/tex] gets larger since we are dealing with a cubic root.
- For negative [tex]\( x \)[/tex] values, [tex]\( y \)[/tex] decreases.
### Final Graph
Here is a rough sketch of how the graph of [tex]\( y = \sqrt[3]{2x} \)[/tex] would look:
1. Start at (0, 0).
2. Pass through points such as (1, 1.26) and (8, 2.52).
3. Mirror these points to the negative side for the symmetry, e.g., (-1, -1.26) and (-8, -2.52).
By following these steps, you'll have an accurate representation of [tex]\( y = \sqrt[3]{2x} \)[/tex]. Remember that the graph is smooth and should not have any sharp corners or discontinuities.
