For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 3 and 4.
y = x³

Verified step by step guidance

1. Understand the given equation: \(y = x^3\) represents a cubic function where the output \(y\) is the cube of the input \(x\). This means for any value of \(x\), \(y\) is calculated by multiplying \(x\) by itself three times.
2. Create a table of values by choosing at least three different values for \(x\). For each chosen \(x\), calculate \(y\) using the formula \(y = x^3\). For example, select values such as \(x = -2\), \(x = 0\), and \(x = 2\).
3. Calculate the corresponding \(y\) values for each \(x\): For \(x = -2\), compute \(y = (-2)^3\); for \(x = 0\), compute \(y = 0^3\); and for \(x = 2\), compute \(y = 2^3\). These ordered pairs \((x, y)\) will be points on the graph.
4. Organize these ordered pairs into a table format with two columns labeled \(x\) and \(y\), listing each pair as a solution to the equation.
5. To graph the equation, plot each ordered pair from the table on the coordinate plane. Connect the points smoothly, noting that the graph of \(y = x^3\) is a curve that passes through the origin and has an S-shape, increasing steeply for positive \(x\) and decreasing steeply for negative \(x\).

Key Concepts

Understanding the Equation y = x³

The equation y = x³ represents a cubic function where the output y is the cube of the input x. This function is continuous and smooth, with an inflection point at the origin (0,0). It is important to recognize how values of x affect y, especially for positive and negative inputs.

Creating Ordered Pairs from a Function

To find ordered pairs (x, y) that satisfy y = x³, select values for x and compute y by cubing those values. For example, if x = 2, then y = 2³ = 8, giving the pair (2, 8). This process helps in plotting points on the graph and understanding the function's behavior.

Graphing Cubic Functions

Graphing y = x³ involves plotting the ordered pairs on a coordinate plane and connecting them smoothly. The graph passes through the origin and extends infinitely in both directions, showing symmetry about the origin (odd function). Recognizing the shape helps visualize growth and decline rates.