Sure, let's analyze the given function [tex]\( y = \frac{1}{4} x^3 \)[/tex]. We'll examine its key properties and its behavior at specific points.
1. Graph Intersection Points:
- Origin: Let's first determine if the graph of this function intersects the x-axis and y-axis at the origin.
- At the origin, [tex]\( x = 0 \)[/tex]:
[tex]\[
y = \frac{1}{4} \cdot 0^3 = 0
\][/tex]
Therefore, the function intersects both the x-axis and y-axis at the origin (0, 0).
- x-intercept: The x-intercept is where [tex]\( y = 0 \)[/tex].
[tex]\[
0 = \frac{1}{4} x^3
\][/tex]
Solving for [tex]\( x \)[/tex],
[tex]\[
x^3 = 0 \implies x = 0
\][/tex]
The x-intercept is at (0, 0).
- y-intercept: The y-intercept is where [tex]\( x = 0 \)[/tex].
[tex]\[
y = \frac{1}{4} \cdot 0^3 = 0
\][/tex]
So, the y-intercept is also at (0, 0).
2. Behavior of the Function:
Next, we need to understand the behavior of the function by calculating some specific values of [tex]\( y \)[/tex] for given [tex]\( x \)[/tex] values.
- For [tex]\( x = -2 \)[/tex],
[tex]\[
y = \frac{1}{4} (-2)^3 = \frac{1}{4} \cdot (-8) = -2
\][/tex]
- For [tex]\( x = -1 \)[/tex],
[tex]\[
y = \frac{1}{4} (-1)^3 = \frac{1}{4} \cdot (-1) = -0.25
\][/tex]
- For [tex]\( x = 1 \)[/tex],
[tex]\[
y = \frac{1}{4} 1^3 = \frac{1}{4} \cdot 1 = 0.25
\][/tex]
- For [tex]\( x = 2 \)[/tex],
[tex]\[
y = \frac{1}{4} 2^3 = \frac{1}{4} \cdot 8 = 2
\][/tex]
The points calculated [tex]\( (-2, -2), (-1, -0.25), (1, 0.25), \)[/tex] and [tex]\( (2, 2) \)[/tex] provide a good overview of the function's behavior:
- For negative [tex]\( x \)[/tex] values, the function produces negative [tex]\( y \)[/tex] values.
- For positive [tex]\( x \)[/tex] values, the function produces positive [tex]\( y \)[/tex] values.
- The function is symmetric about the origin due to the odd power of [tex]\( x \)[/tex].
The function [tex]\( y = \frac{1}{4} x^3 \)[/tex]:
- Passes through the origin (0, 0).
- Has its x-intercept at (0, 0).
- Has its y-intercept at (0, 0).
- Exhibits cubic growth and decay, reflected in the values for specific points as calculated.
This set of points and behavior offers a clear picture of how the function behaves across different intervals of [tex]\( x \)[/tex].
