Problem 30 - Thinking Mathematically

Question

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x^{3}-1\)

Step-by-Step Solution

Answer

After substituting each \(x\) value from the range (-3 to 3) into the equation and generating the corresponding \(y\) values, a table of values showing how \(x\) influences \(y\) is created. The points from the table are then plotted on a graph, and a smooth curve passing through these points is drawn, resulting in the graph of the equation \(y=x^{3}-1\).

Step 1: Calculation of \(y\)'s values

Start by substituting each value of \(x\) given in the range (-3 to 3) into the equation \(y=x^{3}-1\) to find the corresponding \(y\) values.

Step 2: Table of values

Generate a table of values from the calculated points. This will help to visualize the influence of \(x\) on \(y\). For example, when \(x=-3\), \(y=(-3)^{3}-1=-28\), when \(x=-2\), \(y=(-2)^{3}-1=-9\), and so on to \(x=3\), \(y=(3)^{3}-1=26\).

Step 3: Plot the graph

Finally, plot the points on a suitable graph scale. Remember to label the x-axis and y-axis accordingly. Plot the points generated from the table of values on a graph, then draw a smooth curve passing through these points.

Key Concepts

Polynomial Functions

Cubic functions are a type of polynomial function. In a cubic function, the highest degree of the variable is three. For example, in the function \( y = x^3 - 1 \), the term \( x^3 \) is the cube that denotes the function as cubic.
This function is expressed in the form \( ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a eq 0 \). Each term contributes to the overall shape of the graph, such as the direction and steepness of the curve.
Understanding polynomial functions helps in identifying patterns in graphs. Cubic functions, in particular, can have one to three real roots and take the familiar S-shape or backward S-shape. These curves have varied applications, from physics to economics, making them valuable in the study of mathematics.

Coordinate Graphing

Graphing is a method of visually representing relationships between variables in a function. It helps us understand how changes in one variable affect another. With coordinate graphing, we use a grid to plot points and then connect them to see the shape of the graph.
To graph the equation \( y = x^3 - 1 \), you first calculate several coordinate points (\(x, y\)) that satisfy the function. To do this effectively, choose integers for \( x \) within a specific range, such as from -3 to 3, as suggested in the exercise.
Here is a simple process to plot the graph:

• Label your axes: typically, the horizontal axis is the x-axis and the vertical axis is the y-axis.
• Mark each calculated point on your graph, such as (-3, -28) or (0, -1).
• Finally, connect these points smoothly to form the curve of the cubic function.

Graphing is an excellent way to understand functions and their behavior, providing insights into how they change across different values of \( x \).

Table of Values

Creating a table of values is a fundamental step in graphing a function. It organizes the input and output values of a function, providing clarity for both calculation and plotting purposes.
For the cubic function \( y = x^3 - 1 \), compute \( y \) for different values of \( x \) within the range from -3 to 3. These values form the basis of your table.

• When \( x = -3 \), \( y = (-3)^3 - 1 = -28 \).
• When \( x = -2 \), \( y = (-2)^3 - 1 = -9 \).
• When \( x = -1 \), \( y = (-1)^3 - 1 = -2 \).
• Continue this process until \( x = 3 \), where \( y = 26 \).

Once completed, the table reflects the relationship between \( x \) and \( y \) values, preparing these coordinates for plotting on the coordinate plane. This visual representation on a graph helps depict how \( y \) varies with \( x \) in the equation, making it easier to envision the curve's behavior.