Problem 9 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. $$ g(x)=x^{3}-8 $$

Step-by-Step Solution

Answer

Plot points for x values, draw a smooth cubic curve through them.

Step 1: Choose Values for x

Select a few values for \( x \) to plug into the function. Common choices are to use both negative and positive integers, and zero. This helps in visualizing different parts of the curve. Let's choose \( x = -2, -1, 0, 1, 2 \).

Step 2: Calculate Corresponding g(x) Values

Plug the chosen \( x \) values into the function \( g(x) = x^3 - 8 \) to find the corresponding \( g(x) \) values.\\( g(-2) = (-2)^3 - 8 = -8 - 8 = -16 \)\\( g(-1) = (-1)^3 - 8 = -1 - 8 = -9 \)\\( g(0) = (0)^3 - 8 = 0 - 8 = -8 \)\\( g(1) = (1)^3 - 8 = 1 - 8 = -7 \)\\( g(2) = (2)^3 - 8 = 8 - 8 = 0 \)

Step 3: Create a Table of Values

Organize the calculated \( x \) and \( g(x) \) values into a table for clarity.\\[\begin{array}{c|c}x & g(x) \\hline-2 & -16 \-1 & -9 \0 & -8 \1 & -7 \2 & 0 \\end{array}\]

Step 4: Plot the Points on a Graph

Plot the points \((-2, -16), (-1, -9), (0, -8), (1, -7), (2, 0)\) on the coordinate plane. Each point is represented as a dot on the plane.

Step 5: Sketch the Graph

Draw a smooth curve through the plotted points. Since the function \( g(x) = x^3 - 8 \) is a cubic function, expect the curve to have an S-shape. It should decrease towards negative infinity on the left and increase towards positive infinity on the right.

Key Concepts

Cubic Functions

A cubic function is a polynomial function of degree three. The general form is given as \( f(x) = ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants and \( a eq 0 \). The defining characteristic of a cubic function is its highest exponent, which is three, indicating the degree of the polynomial. Cubic functions can have various shapes, but they typically appear as an "S-shaped" curve. This is due to the nature of the cubic term, which dominates the behavior of the function at extreme values of \( x \). The major characteristics include:

• Direction: The curve can increase or decrease at both ends, creating a wave-like pattern.
• Turning Points: Cubic functions can have up to two turning points where the function changes direction.
• Inflection Point: They usually have one inflection point—a spot where the curvature changes from concave up to concave down or vice versa.

Overall, these properties make cubic functions both fun and challenging to graph.

Table of Values

Creating a table of values is a helpful strategy for understanding how a function behaves over a specific range of \( x \) values. By choosing specific points, you can see how the function transitions and starts to take on its curve shape. To start, you select several \( x \) values that span both negative and positive numbers, including zero. This range allows you to observe different aspects of the graph:

• Choose x-values: Opt for integer values unless the problem requires otherwise. For our exercise: \( x = -2, -1, 0, 1, 2 \).
• Calculate y-values: Substitute these \( x \) values into the function to find corresponding \( g(x) \) values like \(-16, -9, -8, -7, 0\).

The resulting table provides a clear map for plotting the function accurately on a graph. It acts as a stepping stone to visualize the cubic function's overall behavior.

Function Graphing

Graphing functions is an essential math skill that involves plotting points on a coordinate plane. You begin by using the table of values you created, which guides the placement of points on the graph. Here is a quick guide to function graphing:

• Plot Points: Take the \((x, g(x))\) pairs from the table and mark these coordinates on the graph. For example, the points \((-2, -16), (-1, -9), (0, -8), (1, -7), (2, 0)\).
• Connect the Dots: Use a smooth and continuous curve to join the plotted points. Cubic graphs should gently swerve in an S-shape.

Proper graphing allows you to detect the function's critical features like intercepts, turning points, and asymptotic behavior, presenting a visual representation of the function's algebraic equation.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we plot functions, consisting of two perpendicular axes, \( x \) (horizontal) and \( y \) (vertical). Each point on this plane is defined by a pair of numeric coordinates \((x, y)\), which dictates its position relative to these axes. Here are some key facets of the coordinate plane:

• Axes: The \( x \)-axis and \( y \)-axis split the plane into four quadrants where points can be plotted.
• Quadrants: The plane is divided clockwise into four areas. Points in the top right have both positive \( x \) and \( y \) values, while those below the \( x \)-axis have negative \( y \) values.

A coordinate plane is essential for graphing any function as it provides the structure and reference needed to accurately display mathematical relationships. Understanding this space helps you translate function points into visual graphs.