Problem 28 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

Filled table of x and y values: \[x = -3, y = -28; x = -2, y = -9; x = -1, y = -2; x = 0, y = -1; x = +1, y = 0; x = +2, y = 7; x = +3, y = 26.\] The plotted points will form an 'S' shape curve when graphed.

Step 1: Understand the function

It's crucial in beginning to recognize that \(y = x^3 - 1\) is a cubic function. Cubic functions usually produce curved graphs, unlike linear functions which create straight line graphs. The equation means that having chosen a value for x, you raise it to the power of three and subtract 1 to attain the y value.

Step 2: Generate y-values

Substitute each of the given x-values into the equation, \(x^3 - 1\), to get the corresponding y-values. Using these pairs of x and y values, a table can be created. For instance, when \(x = -3)\, the equation transforms to \(y = (-3)^3 - 1 = -28\). Proceeding like this with the rest of the x-values, a table of x and y values is formed ready for graphing.

Step 3: Plot the data points

Using the table of values, each pair of x and y are plotted as data points on a graph. This step serves as a means to visualise the data before drawing the curve.

Step 4: Draw the curve

Drawing the curve starts from the first left value on the graph, keeping in mind that cubic functions result in a curved line. Each plotted point will be linked by this curve and it should pass smoothly from one point to another, creating a visual illustration of the equation.

Key Concepts

Graphing

Graphing is a great way to visualize mathematical relationships, especially with equations like the cubic function. A cubic function such as \(y = x^3 - 1\) results in a graph that isn't a straight line but rather a curve.
To graph a function, you need to plot points in a coordinate plane. Each point represents a specific value of \(x\) and its corresponding value of \(y\). For a cubic equation, like the one we are considering, you'll notice that the graph will have a unique shape, often resembling an elongated 'S' curve.
This curve arises because the output values, or \(y\) values, change in a non-linear fashion as \(x\) changes. By graphing such equations, students can better understand how the rate of change varies across different intervals.

Coordinate Pairs

Coordinate pairs are like the building blocks of graphing functions. Each pair consists of an \( x \) value and a corresponding \( y \) value, adhering to the rule of the given function. In our specific exercise, you would have pairs like \((-3, -28)\), where \(-3\) is the \(x\) value and \(-28\) is the \(y\) value, calculated using the equation \(y = x^3 - 1\).

• The coordinate pair \((x, y)\) describes precisely where to place a point on the graph.
• Each pair corresponds to a location on the graph, helping students see how the function translates from an algebraic equation to a visual representation.

When plotting these points, they should be evenly spaced to represent the changes of \(y\) with varying \(x\). Once all points are plotted, connecting these points with a smooth curve serves to highlight the overall behavior of the function.

Polynomial Equations

Polynomial equations are expressions consisting of variables and coefficients, involving terms of varying powers combined by addition, subtraction, or multiplication. Cubic equations, like the one in our example, are a specific type of polynomial where the highest degree is 3.

• The general form of a cubic polynomial is \(ax^3 + bx^2 + cx + d\). For our equation, that's simplified to \(x^3 - 1\) with a single \(x^3\) term.
• Cubic functions illustrate more complex behaviors than linear or quadratic polynomials, allowing for an in-depth exploration of mathematical relationships.

Understanding polynomial equations and their graphs can help in visualizing how solutions to these polynomials behave. Higher-degree polynomials produce more complex curves, which is essential in advanced mathematics, physics, and engineering.