Question

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ g(u)=\frac{u^{3}}{8} $$

Short answer

The function is odd.

Step-by-step solution

Define Even and Odd Functions

A function is even if \( f(-x) = f(x) \) for all \( x \), which means the graph is symmetrical about the y-axis. A function is odd if \( f(-x) = -f(x) \), meaning the graph is symmetrical about the origin. If neither condition is met, the function is neither even nor odd.

Substitute to Test Evenness

Substitute \( -u \) into the function: \( g(-u) = \frac{(-u)^3}{8} = \frac{-u^3}{8} \). Now compare it with \( g(u) = \frac{u^3}{8} \). Since \( g(-u) eq g(u) \), the function is not even.

Substitute to Test Oddness

Check if \( g(-u) = -g(u) \): We have \( g(-u) = \frac{-u^3}{8} \) and \(-g(u) = -\frac{u^3}{8} = \frac{-u^3}{8} \). Since \( g(-u) = -g(u) \), the function is odd.

Sketch the Graph

Since the function is odd, the graph will be symmetric about the origin. Plot several key points: at \( u = 0 \), \( g(0) = 0 \); at \( u = 2 \), \( g(2) = \frac{8}{8} = 1 \), and for \( u = -2 \), \( g(-2) = \frac{-8}{8} = -1 \). The graph will pass through (0,0), (2,1), and (-2,-1) displaying a cubic shape extending infinitely in opposite quadrants.

Key concepts

Graph Symmetry

Graph symmetry refers to a graphical property where parts of the graph mirror each other across a line or point. Symmetry can help us understand the behavior of a function visually.
For even functions, the symmetry is about the y-axis. This means if you fold the graph along the y-axis, both halves will overlap perfectly. Mathematically, this condition is checked by confirming if the function satisfies the equation: \[ f(-x) = f(x) \] An example of an even function is a parabola, like \( f(x) = x^2 \).
For odd functions, the symmetry is about the origin. Here, if you rotate the graph 180 degrees around the origin, it looks the same. This is tested with: \[ f(-x) = -f(x) \] The function \( g(u) = \frac{u^3}{8} \) is odd, as shown by checking the condition. This symmetry simplifies visualization, knowing the graph's shape in one quadrant tells us the shape in the opposite quadrant should be inverted.
Identifying symmetry helps not just in sketching but also in predicting function behaviors across a real-world context.

Function Analysis

Function analysis involves examining functions to understand their behavior, including their symmetry, domain, range, and type (such as even/odd). This process helps in predicting a graph's characteristics without plotting every point.
To analyze our function \( g(u)=\frac{u^3}{8} \), we first determine its symmetry. By substituting \(-u\) into the function, we test for evenness: \[ g(-u) = \frac{(-u)^3}{8} = \frac{-u^3}{8} \] This comparison with \( g(u) \) shows it is not even because they don't match.
Next, we check for oddness by evaluating whether: \[ g(-u) = -g(u) \] Indeed, both equal \( \frac{-u^3}{8} \), confirming it is odd.
Knowing this guides us in sketching the graph, ensuring we represent a correct shape and making it easier to predict other features like intercepts and potential stretching/shrinking based on transformations.
Mastering function analysis allows us to unlock deeper insights into mathematical models and real-world applications.

Cubic Functions

Cubic functions are polynomials of degree three, typically expressed in the form \( f(x) = ax^3 + bx^2 + cx + d \). They can exhibit interesting graph shapes that are not linear or quadratic. The function \( g(u)=\frac{u^3}{8} \) is a cubic function with a single term, making it a simpler case to analyze.
The graph of a cubic function generally has one or more turning points, but a simple cube like \( g(u) \) just passes through the origin, creating an S-shape. Cubic graphs extend infinitely in both directions, often crossing the x-axis up to three times in more complex forms.
Key characteristics of cubic functions include:

• Intercepts: A cubic function typically crosses the y-axis once, at its constant term.
• End behavior: As the input value increases or decreases, the output will continue in that respective upward or downward direction, based on the leading coefficient's sign.
• Roots and factors: It can have up to three real roots, which are the x-intercepts where the graph crosses the x-axis.

Understanding these properties can help us anticipate the graph's appearance and behavior, making it easier to work with complex applications in physics and engineering where cubic models often arise.