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 = x²

Verified step by step guidance

1. Step 1: Understand the equation given, which is \(y = x^{2}\). This is a quadratic function where \(y\) is the square of \(x\). The graph of this equation is a parabola opening upwards.
2. Step 2: To create a table of ordered pairs, choose at least three values for \(x\). For example, select \(x = -1\), \(x = 0\), and \(x = 1\).
3. Step 3: Calculate the corresponding \(y\) values by substituting each \(x\) into the equation \(y = x^{2}\). For \(x = -1\), compute \(y = (-1)^{2}\); for \(x = 0\), compute \(y = 0^{2}\); and for \(x = 1\), compute \(y = 1^{2}\).
4. Step 4: Write the ordered pairs as \((x, y)\) using the values found in Step 3. These pairs will be points on the graph of the equation.
5. Step 5: To graph the equation, plot the ordered pairs on the coordinate plane and draw a smooth curve through these points forming a parabola opening upwards. You can also choose additional \(x\) values to get more points for a more accurate graph.

Key Concepts

Understanding the Equation y = x²

The equation y = x² represents a quadratic function where the output y is the square of the input x. This creates a parabola opening upwards, symmetric about the y-axis. Recognizing this form helps in predicting the shape and behavior of the graph.

Creating Ordered Pairs from an Equation

To find ordered pairs (x, y) that satisfy y = x², select values for x and compute y by squaring x. This process generates points on the graph, which can be tabulated to visualize the function's behavior and assist in plotting.

Graphing Quadratic Functions

Graphing y = x² involves plotting the ordered pairs on a coordinate plane and connecting them smoothly to form a parabola. Understanding key features like the vertex at (0,0) and symmetry helps in accurately sketching the curve.