Problem 88 - Algebra and Trigonometry

Question

\(87-88\) a Sketch the graph of each function. $$ (a)f(x)=x^{3} \quad \text { (b) } g(x)=\left|x^{3}\right| $$

Step-by-Step Solution

Answer

Sketch the 'S' shape for \( f(x) = x^3 \) and a 'U' shape flipped for negative \( x \) for \( g(x) = |x^3| \).

Step 1: Understanding the Function f(x)

The function \( f(x) = x^3 \) represents a cubic function. Cubic functions are continuous and smooth, meaning they have no breaks, holes, or sharp corners. This function has certain key features: it passes through the origin (0, 0), it increases to infinity as \( x \) approaches positive infinity, and decreases to negative infinity as \( x \) approaches negative infinity. Additionally, it is odd, meaning that \( f(-x) = -f(x) \).

Step 2: Sketching f(x) = x^3

Begin plotting key points such as (0,0), (1,1), and (-1,-1). Since this is a cubic function, you can expect the curve to increase sharply as you move to larger positive values of \(x\) and decrease sharply for negative \(x\). The plot will have a sort of 'S' shape, smoothly transitioning through the origin without any sharp turns, and symmetry with respect to the origin.

Step 3: Understanding the Function g(x)

The function \( g(x) = |x^3| \) represents the absolute value of a cubic function. This means it will always yield a non-negative result. While it has the same absolute numerical values as the cubic function, it differs around the origin. Specifically, for negative values of \( x \), \( g(x) \) reflects the negative part of \( f(x) \) to the positive side, resulting in a 'U' shape near the origin.

Step 4: Sketching g(x) = |x^3|

Plot key points such as (0,0), (1,1), and (-1,1). Unlike \( f(x) = x^3 \), the graph of \( g(x) = |x^3| \) will not pass below the x-axis. For \( x > 0 \), \( g(x) \) is identical to \( f(x) \). For \( x < 0 \), reflect the \( f(x) \) part across the x-axis (making \( g(x) = -f(x) \)), which results in a slope that creates a sharp 'V' shape at the origin.

Key Concepts

Cubic Functions

Cubic functions have the general form of \( f(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants and \( a eq 0 \). One of the simplest examples of a cubic function is \( f(x) = x^3 \). These functions are known for their S-shaped curves, which are symmetric about the origin.
Key features of cubic functions include:

• Smooth, continuous curves: No breaks or sharp turns exist in the function's graph.
• Pass through the origin: The point (0,0) is a notable feature for \( f(x) = x^3 \).
• End behavior: As \( x \) increases, \( f(x) \) also increases sharply; as \( x \) decreases, \( f(x) \) decreases sharply.
• Odd symmetry: If you change \( x \) to \(-x\), the function shows \( f(-x) = -f(x) \).

These characteristics make cubic functions a topic of interest when studying different types of polynomial functions. They exhibit unique turning points and inflection points that provide important analytical insights.

Absolute Value Functions

Absolute value functions represent the distance of a number from zero on the number line, and are always non-negative. When applied to cubic functions such as \( g(x) = |x^3| \), they transform the original cubic function's graph.
For the function \( g(x) = |x^3| \):

• Non-negativity: The graph will never go below the x-axis, as absolute values are always positive or zero.
• Reflection for negative \( x \): At points where \( x < 0 \), the graph reflects the negative parts of the cubic function, making them positive.
• Symmetrical 'V' shape: This results in a 'V' shaped symmetry close to the origin, not the S shape seen in \( f(x) = x^3 \).

Absolute value functions are often used to understand magnitude without considering direction, which can highlight symmetry in unconventional forms.

Function Symmetry

Function symmetry can make it easier to understand the behavior of a graph. Symmetrical functions, such as odd, even or those showing line symmetry, often have predictable patterns in their graphs.
- **Odd Symmetry:** Odd functions satisfy \( f(-x) = -f(x) \), meaning one half of the graph is a reflection through the origin to the other half. An example is the cubic function \( f(x) = x^3 \). This symmetry suggests that if a point \((a, b)\) is on the graph, \((-a, -b)\) will also be on it.
- **Even Symmetry:** Unlike odd functions, even functions have symmetry about the y-axis. A function is even if \( f(-x) = f(x) \). Absolute value functions, when applied as in \( g(x) = |x^3| \), reveal an even-like modified symmetry visible in their graph.
This intrinsic property helps mathematicians and students alike by simplifying the graphing process and offering visual cues to better understand function behavior.

Graph Transformations

Graph transformations are changes made to the basic graph of a function, which alter its position or shape. Understanding these transformations is key to mastering graph plotting and analysis.
- **Vertical shifts:** Moving the graph up or down involves adding or subtracting a constant \( c \) to/from the function, \( f(x) + c \).
- **Horizontal shifts:** Adjusting the graph left or right occurs by adding or subtracting a constant inside the function's argument, like \( f(x - c) \).
- **Reflections:** To reflect a graph over the x-axis, multiply the entire function by \(-1\). For example, turning \( f(x) \) into \( -f(x) \) flips it vertically. For a y-axis reflection, modify \( x \) so \( f(-x) \) reflects it horizontally.
- **Stretches and Compressions:** Multiplying or dividing the function by a factor will stretch (if the factor is greater than 1) or compress (if between 0 and 1) the graph vertically.

• For instance, the \( g(x) = |x^3| \) transformation takes the cubic graph and prevents it from dipping below the x-axis, reflecting negative values upwards for new graph forms.

These transformations are powerful tools for customizing graphs to fit specific mathematical needs or demonstrate particular properties visually.