Answer :
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.
