For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. y=x²

Verified step by step guidance

1. Identify the given equation: \(y = x^2\). This is a quadratic function where \(y\) is the square of \(x\).
2. Choose at least three values for \(x\). For example, select \(x = -1\), \(x = 0\), and \(x = 2\) to find corresponding \(y\) values.
3. Calculate the \(y\) values by substituting each chosen \(x\) into the equation \(y = x^2\). For instance, when \(x = -1\), compute \(y = (-1)^2\).
4. Create a table of ordered pairs \((x, y)\) using the values found. For example, the pairs might look like \((-1, 1)\), \((0, 0)\), and \((2, 4)\).
5. To graph the equation, plot each ordered pair on the coordinate plane and connect the points with a smooth curve forming a parabola opening upwards.

Key Concepts

Ordered Pairs as Solutions to Equations

An ordered pair (x, y) represents a solution to an equation if substituting x into the equation yields the corresponding y value. For example, in y = x², if x = 2, then y = 4, so (2, 4) is a solution. Creating a table of such pairs helps visualize the relationship between variables.

Quadratic Functions and Their Graphs

A quadratic function has the form y = ax² + bx + c, where the graph is a parabola. For y = x², the parabola opens upward with its vertex at the origin (0,0). Understanding this shape helps in sketching the graph accurately based on the ordered pairs.

Plotting Points and Graphing Equations

Graphing involves plotting ordered pairs on the coordinate plane and connecting them smoothly. For y = x², plotting points like (-1,1), (0,0), and (1,1) reveals the curve's shape. This visual representation aids in understanding the function's behavior.