Problem 28 - College Algebra

Question

Graph each equation in Exercises \(13-28 .\) Let \(x=-3,-2,-1\) \(0,1,2,\) and 3. $$y=x^{3}-1$$

Step-by-Step Solution

Answer

The graph of \(y=x^{3}-1\) will be a curve passing through the points corresponding to the pairs of x- and y-values. Note that for every x-value, the y-value is calculated by cubing the x-value and subtracting 1.

Step 1: Understanding the equation

The given equation is \(y=x^{3}-1\). Here, x is the input, and y is the output. For any x-value, y is determined by cubing the x-value and subtracting 1.

Step 2: Substituting the x-values

Substitute each of the x-values (-3, -2, -1, 0, 1, 2, 3) into the equation to determine the corresponding y-values.

Step 3: Graphing the equation

Plot each of the pairs (x, y) on a graph. This will give a curve, which is the graph of the equation \(y=x^{3}-1\).

Key Concepts

Cubic Equations

Cubic equations are mathematical expressions where the highest exponent of the variable is three. In simpler terms, it means that the equation is based on the cube of a variable, typically denoted as \(x\). A general form of a cubic equation is \(y = ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants, but in most beginner exercises, you will often see them simplified, like the equation \(y = x^3 - 1\).

• These equations can produce various types of graphs and complex patterns.
• The key feature of a cubic graph is its characteristic "S" shape, but this can vary based on the coefficients.

Cubic equations are fundamental in learning how different algebraic expressions influence graph shapes.Think of a roller-coaster, where the path of the ride can be likened to the curve of a cubic graph.

Plotting Points

To visually represent an equation like \(y = x^3 - 1\), plotting points is an essential step. Every equation has what we call "solutions," which are specific points on the curve corresponding to particular \(x\) values. Here's how to plot points from the equation:

• Choose a set of \(x\)-values— as in this exercise, \(-3, -2, -1, 0, 1, 2, 3\).
• Substitute these \(x\)-values into your cubic equation to calculate the corresponding \(y\)-values.
• Once you've got each \(y\)-value, they are paired with their respective \(x\), forming ordered pairs, such as \((x, y)\).

For example, if \(x = 1\), substituting into the equation gives \(y = 1^3 - 1 = 0\), so you have the point \((1, 0)\).This method helps us systematically create a blueprint to sketch the curve on a graph.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which you can plot points, lines, and curves. It consists of an \(x\)-axis (horizontal) and a \(y\)-axis (vertical), which intersect at a point called the origin, labeled \( (0,0) \). Here's how it helps with graphing:

• Each point on the plane is defined by an \(x\)-coordinate and a \(y\)-coordinate, creating a grid that allows you to precisely locate points such as \((-3, 26)\) or \((3, 26)\).
• This precise framework allows clear visualization of the relations between numbers and shapes.

When you plot cubic equations on the coordinate plane, the points you plotted begin to form a recognizable curve as you connect them.

Function Graphing

Graphing a function is like sketching a picture that shows the behavior of an equation visually. For a cubic function such as \(y = x^3 - 1\), graphing it turns numerical solutions into a visible and understandable form. The process generally involves:

• Calculating and plotting the points as explained in the previous sections.
• Smoothly connecting these points with a continuous curve.
• Understanding the curve's shape and how it represents the relationship between \(x\) and \(y\).

When graphing, you'll notice that the curve will pass through all your plotted points, representing how the value of \(y\) changes with \(x\).This not only aids in solving algebraic problems, but also enhances understanding of how variables interact geometrically.