Sure! Let's walk through the steps to graph the cubic function [tex]\( y = \frac{3}{4} x^3 \)[/tex] and plot five points where [tex]\( x = 0 \)[/tex] and two points on either side of [tex]\( x = 0 \)[/tex].
Step 1: Understanding the Function
The function we are graphing is:
[tex]\[ y = \frac{3}{4} x^3 \][/tex]
Step 2: Selecting Points
We need to select five points to graph:
1. [tex]\( x = 0 \)[/tex]
2. Two points to the left of [tex]\( x = 0 \)[/tex] (choose [tex]\( x = -1 \)[/tex] and [tex]\( x = -2 \)[/tex])
3. Two points to the right of [tex]\( x = 0 \)[/tex] (choose [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex])
Step 3: Calculating the Corresponding [tex]\( y \)[/tex]-values
For each [tex]\( x \)[/tex]-value, we will calculate the corresponding [tex]\( y \)[/tex]-value using the function [tex]\( y = \frac{3}{4} x^3 \)[/tex]:
1. [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{3}{4} (0)^3 = 0 \][/tex]
Point: (0, 0)
2. [tex]\( x = -1 \)[/tex]:
[tex]\[ y = \frac{3}{4} (-1)^3 = \frac{3}{4} \times (-1) = -\frac{3}{4} \][/tex]
Point: (-1, -0.75)
3. [tex]\( x = -2 \)[/tex]:
[tex]\[ y = \frac{3}{4} (-2)^3 = \frac{3}{4} \times (-8) = -6 \][/tex]
Point: (-2, -6)
4. [tex]\( x = 1 \)[/tex]:
[tex]\[ y = \frac{3}{4} (1)^3 = \frac{3}{4} \][/tex]
Point: (1, 0.75)
5. [tex]\( x = 2 \)[/tex]:
[tex]\[ y = \frac{3}{4} (2)^3 = \frac{3}{4} \times 8 = 6 \][/tex]
Point: (2, 6)
Step 4: Plotting the Points
We will now plot these points on a graph and draw the curve representing the function [tex]\( y = \frac{3}{4} x^3 \)[/tex].
- Plot the points [tex]\((0, 0)\)[/tex], [tex]\((-1, -0.75)\)[/tex], [tex]\((-2, -6)\)[/tex], [tex]\((1, 0.75)\)[/tex], and [tex]\((2, 6)\)[/tex] on a Cartesian plane.
- Draw a smooth curve through these points to represent the cubic function [tex]\( y = \frac{3}{4} x^3 \)[/tex].
Graph:
```
x | y
-----------
-2 | -6
-1 | -0.75
0 | 0
1 | 0.75
2 | 6
```
Visual Representation:
```
^
|
|
|
| (0,0)
|
|
|
+----------------------------------------------------->
-2 -1 0 1 2
```
The cubic function [tex]\( y = \frac{3}{4} x^3 \)[/tex] has the characteristic S-shaped curve, passing through the origin and displaying symmetry with respect to the origin.
