Problem 29 - Thinking Mathematically

Question

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x^{3}\)

Step-by-Step Solution

Answer

The graph of the function \(y=x^3\) is a curve that begins in quadrant 3 and moves into quadrant 1 as \(x\) increases from -3 to 3. It passes through seven points: (-3, -27), (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8), and (3, 27).

Step 1: Creating a Table

The first step is to construct a table of values to represent the co-ordinates of each point in the graph. For the given cubic function \(y = x^3\), we are asked to consider integer values of \(x\) from -3 to 3. That gives us seven points to plot. We evaluate \(y\) by substituting the \(x\) values into the equation \(y = x^3\).

Step 2: Fill in the Table of Values

The table of values now looks like: (-3, -27), (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8), (3, 27). This is done by taking each of the \(x\) values, substituting them into \(y = x^3\), and recording the resulting \(y\) values.

Step 3: Plot the Co-ordinates and Construct the Graph

After acquiring all the points from the table, the final step is to plot each of these points onto the graph. By joining these points, we can visualize the graph of the function \(y=x^3\) within the range of -3 and 3 for \(x\). The graph starts from the third quadrant(-3,-27) and goes into the first quadrant(3,27). Because of the cubic function, the pattern of the graph shows a curve

Key Concepts

Cubic Functions

Cubic functions are mathematical expressions of degree three. They take the form of \( y = ax^3 + bx^2 + cx + d \), where \( a eq 0 \). In the simple cubic function \( y = x^3 \), there are no quadratic (\( x^2 \)), linear (\( x \)) or constant terms (\( d \)).
This specific cubic function demonstrates a symmetric nature, which means it behaves similarly on both the positive and negative side of the graph.
You can recognize cubic functions by their S-shaped curve on a graph. The features of a cubic function include:

• Turning points: A cubic function can have up to two turning points. These are local maximum or minimum points where the graph changes direction.
• Intercepts: This function passes through the origin, \( (0, 0) \), as it has no constant term. The x-intercepts and y-intercepts coincide at this point.

This balanced structure makes cubic functions smooth, with a continuous curve for all \( x \).

Coordinate Geometry

Coordinate geometry involves plotting points, lines, and curves on a plane using pairs of numbers called coordinates. In graphing, coordinates utilize the x-y plane, where any point is specified by its \( x \) and \( y \) value.
The pair \( (x, y) \) indicates the position of a point across the Cartesian coordinate system:

• \( x \): Represents the horizontal position, positive to the right and negative to the left.
• \( y \): Indicates the vertical position, positive upwards and negative downwards.

In cubic functions like \( y = x^3 \), understanding the coordinate geometry helps visualize the relationships between \( x \) and \( y \).
Plotting these points provides a graphical representation of the math equation, helping to see how changes in \( x \) affect \( y \), forming the characteristic shape of the graph.

Graph Plotting

Graph plotting is an essential skill that visualizes mathematical equations effectively. For a function like \( y = x^3 \), plotting the points creates a clear image of how the function behaves and changes.
Steps to plot the graph:

• Start by identifying the points you will plot. From the cubic equation \( y = x^3 \), substitute integral \( x \) values.
• Next, calculate the corresponding \( y \) values to create coordinate pairs, as shown in the solution: (-3, -27), (-2, -8), ..., (3, 27).
• Place each coordinate on the graph, noting where they fall on the x-y plane.
• Finally, draw a smooth curve connecting the points, ensuring the S-shaped curve typical of cubic functions.

This visualization allows one to observe the shape and trajectory of the function, a vital step in analyzing complex functions.

Table of Values

Creating a table of values is a practical approach to systematically determine and visualize coordinates for a function. For the cubic function \( y = x^3 \), constructing a table helps in quickly identifying the exact points to plot on a graph.

• Select integers for \( x \) within the given range (-3 to 3).
• For each value of \( x \), compute \( y \) using the function \( y = x^3 \).
• List each result as an ordered pair \( (x, y) \).

The table for this cubic function would be:
- \((-3, -27)\)
- \((-2, -8)\)
- \((-1, -1)\)
- \((0, 0)\)
- \((1, 1)\)
- \((2, 8)\)
- \((3, 27)\)
This method not only makes plotting straightforward but also reinforces understanding of how equation values translate directly to points on a graph.