IDNLearn.com: Your trusted platform for finding precise and reliable answers. Ask anything and receive comprehensive, well-informed responses from our dedicated team of experts.
Sagot :
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.
### 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.
Your presence in our community is highly appreciated. Keep sharing your insights and solutions. Together, we can build a rich and valuable knowledge resource for everyone. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.
