Problem 69 - Intermediate Algebra

Question

Graph each of the basic functions. $$ h(x)=x^{3} $$

Step-by-Step Solution

Answer

The graph of \( h(x) = x^3 \) is an S-shaped curve passing through the origin with points like \((-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8)\).

Step 1: Understand the Function

The function given is a cubic function, specifically a basic power function of the form \( h(x) = x^3 \). A cubic function is odd and has rotational symmetry about the origin.

Step 2: Identify Key Features

To graph the cubic function \( h(x) = x^3 \), first identify critical points and behavior:- **Intercepts**: The graph intercepts the origin at \((0, 0)\).- **Behavior**: As \(x\) approaches infinity, \(h(x)\) increases to infinity; as \(x\) approaches negative infinity, \(h(x)\) decreases to negative infinity. There is no maximum or minimum point for this function.

Step 3: Calculate Additional Points

To aid in sketching, calculate a few more points:- \( x = -2 \), \( h(-2) = (-2)^3 = -8 \)- \( x = -1 \), \( h(-1) = (-1)^3 = -1 \)- \( x = 1 \), \( h(1) = 1^3 = 1 \)- \( x = 2 \), \( h(2) = 2^3 = 8 \)

Step 4: Plot the Points and Sketch the Curve

Plot these points on a coordinate plane: \((-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8)\). Connect the points with a smooth curve through all the points, maintaining symmetry about the origin. The resulting shape is an 'S'-shaped curve, characteristic of cubic functions.

Key Concepts

Cubic Functions

Cubic functions are a fascinating group of functions that are represented by an equation of the form \( f(x) = ax^3 + bx^2 + cx + d \). In our original exercise, we are tasked with graphing the simplest form of a cubic function: \( h(x) = x^3 \). These functions are essential because they exhibit unique characteristics that differ from linear and quadratic functions.

• Shape: A basic cubic function, like \( x^3 \), has an "S"-shaped curve characteristic.
• Direction: It has continuous growth from negative to positive infinity.
• Odd Function: It is symmetric about the origin, which is a type of symmetry we'll explore further.

Understanding cubic functions can greatly enhance one's grasp of higher-order polynomials, making them a valuable tool in mathematical learning.

Power Functions

Power functions take the form \( f(x) = ax^n \) where \( a \) and \( n \) are constants. Our cubic function \( h(x) = x^3 \) is a type of power function where \( n = 3 \) and \( a = 1 \). Power functions are important as they show how powers dictate the overall shape and behavior of graphs.

• Exponent Influence: The exponent "3" in \( x^3 \) creates the cubic shape, showing how the graph's steepness and curvature change across the x-axis.
• Leading Coefficient: The leading coefficient "1" makes the function's growth rate predictable and consistent.
• Simple Example: Working with \( h(x) = x^3 \) provides a clearer understanding of how variations in \( a \) or \( n \) affect the graph.

As you learn to recognize power functions, you'll see them surface in numerous mathematical applications.

Symmetry

Symmetry in functions is a property that makes graphing simpler and graphs more aesthetically pleasing. Cubic functions like \( h(x) = x^3 \) have a special kind of symmetry called rotational symmetry. Specifically, they are symmetric with respect to the origin.

• Origin Symmetry: A graph has origin symmetry if rotating it 180 degrees around the origin results in the same graph.
• Odd Function: Cubic functions are odd, which means \( f(-x) = -f(x) \). This indicates that they possess origin symmetry.
• Graphical Implication: When sketching, if you know the setup of one half of the graph, you can easily deduce the other half due to this symmetry.

Recognizing symmetry can save time and improve accuracy when plotting functions.

Intercepts

Intercepts are key features on any graph, indicating where the function crosses the axes. For cubic functions like \( h(x) = x^3 \), intercepts are straightforward but crucial to understand.

• X-Intercept: The point(s) where the graph crosses the x-axis. Here, it's at \((0,0)\) because \( h(x) = 0 \) when \( x = 0 \).
• Y-Intercept: The point where the graph crosses the y-axis. In this case, the y-intercept is also \((0,0)\).
• Importance: Knowing intercepts helps quickly sketch the function and assess its initial direction.

Grasping intercepts is fundamental as it provides a "starting point" for understanding the behavior of more complex graphs.