At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Certainly! Let's graph the function [tex]\( y = -\frac{1}{2} x^3 \)[/tex] and plot five specific points on the graph. We'll be plotting one point with [tex]\( x = 0 \)[/tex], two points with negative [tex]\( x \)[/tex]-values, and two points with positive [tex]\( x \)[/tex]-values.
### Step-by-Step Solution:
1. Calculate the y-values for each given x-value:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (0)^3 = -0.0 \][/tex]
So, the point is [tex]\( (0, -0.0) \)[/tex].
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (-2)^3 = -\frac{1}{2} \cdot (-8) = 4.0 \][/tex]
So, the point is [tex]\( (-2, 4.0) \)[/tex].
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (-1)^3 = -\frac{1}{2} \cdot (-1) = 0.5 \][/tex]
So, the point is [tex]\( (-1, 0.5) \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (1)^3 = -\frac{1}{2} \cdot 1 = -0.5 \][/tex]
So, the point is [tex]\( (1, -0.5) \)[/tex].
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (2)^3 = -\frac{1}{2} \cdot 8 = -4.0 \][/tex]
So, the point is [tex]\( (2, -4.0) \)[/tex].
2. Summarize the points:
- [tex]\( (0, -0.0) \)[/tex]
- [tex]\( (-2, 4.0) \)[/tex]
- [tex]\( (-1, 0.5) \)[/tex]
- [tex]\( (1, -0.5) \)[/tex]
- [tex]\( (2, -4.0) \)[/tex]
3. Plot the points on the graph:
To graph the function, plot the five points on a coordinate plane:
- Point 1: [tex]\( (0, -0.0) \)[/tex]
- Point 2: [tex]\( (-2, 4.0) \)[/tex]
- Point 3: [tex]\( (-1, 0.5) \)[/tex]
- Point 4: [tex]\( (1, -0.5) \)[/tex]
- Point 5: [tex]\( (2, -4.0) \)[/tex]
4. Draw the curve:
Connect the points smoothly to represent the function [tex]\( y = -\frac{1}{2} x^3 \)[/tex]. The graph should show the characteristic curve of a cubic function, with a point of inflection at the origin where the concavity changes.
5. Graph Characteristics:
- The curve will be descending as [tex]\( x \)[/tex] moves from negative to positive values.
- It will pass through the origin at [tex]\( (0, -0.0) \)[/tex].
- The curve will be symmetrical about the origin.
- The function [tex]\( y = -\frac{1}{2} x^3 \)[/tex] is odd, and thus the graph exhibits rotational symmetry about the origin, meaning [tex]\( f(-x) = -f(x) \)[/tex].
By following these steps, you've successfully graphed the function [tex]\( y = -\frac{1}{2} x^3 \)[/tex] with the given five points.
### Step-by-Step Solution:
1. Calculate the y-values for each given x-value:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (0)^3 = -0.0 \][/tex]
So, the point is [tex]\( (0, -0.0) \)[/tex].
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (-2)^3 = -\frac{1}{2} \cdot (-8) = 4.0 \][/tex]
So, the point is [tex]\( (-2, 4.0) \)[/tex].
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (-1)^3 = -\frac{1}{2} \cdot (-1) = 0.5 \][/tex]
So, the point is [tex]\( (-1, 0.5) \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (1)^3 = -\frac{1}{2} \cdot 1 = -0.5 \][/tex]
So, the point is [tex]\( (1, -0.5) \)[/tex].
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \cdot (2)^3 = -\frac{1}{2} \cdot 8 = -4.0 \][/tex]
So, the point is [tex]\( (2, -4.0) \)[/tex].
2. Summarize the points:
- [tex]\( (0, -0.0) \)[/tex]
- [tex]\( (-2, 4.0) \)[/tex]
- [tex]\( (-1, 0.5) \)[/tex]
- [tex]\( (1, -0.5) \)[/tex]
- [tex]\( (2, -4.0) \)[/tex]
3. Plot the points on the graph:
To graph the function, plot the five points on a coordinate plane:
- Point 1: [tex]\( (0, -0.0) \)[/tex]
- Point 2: [tex]\( (-2, 4.0) \)[/tex]
- Point 3: [tex]\( (-1, 0.5) \)[/tex]
- Point 4: [tex]\( (1, -0.5) \)[/tex]
- Point 5: [tex]\( (2, -4.0) \)[/tex]
4. Draw the curve:
Connect the points smoothly to represent the function [tex]\( y = -\frac{1}{2} x^3 \)[/tex]. The graph should show the characteristic curve of a cubic function, with a point of inflection at the origin where the concavity changes.
5. Graph Characteristics:
- The curve will be descending as [tex]\( x \)[/tex] moves from negative to positive values.
- It will pass through the origin at [tex]\( (0, -0.0) \)[/tex].
- The curve will be symmetrical about the origin.
- The function [tex]\( y = -\frac{1}{2} x^3 \)[/tex] is odd, and thus the graph exhibits rotational symmetry about the origin, meaning [tex]\( f(-x) = -f(x) \)[/tex].
By following these steps, you've successfully graphed the function [tex]\( y = -\frac{1}{2} x^3 \)[/tex] with the given five points.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
