For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 7 and 8. y=x^2

Verified step by step guidance

1. Step 1: Understand the equation. The given equation is a quadratic equation in the form of y = x^2. This means for every x-value, y is the square of x.
2. Step 2: Choose three x-values to find corresponding y-values. For simplicity, let's choose x = -1, x = 0, and x = 1.
3. Step 3: Calculate the y-values for each chosen x-value. For x = -1, y = (-1)^2 = 1. For x = 0, y = 0^2 = 0. For x = 1, y = 1^2 = 1.
4. Step 4: Create a table of ordered pairs. The ordered pairs are (-1, 1), (0, 0), and (1, 1).
5. Step 5: Graph the equation. Plot the ordered pairs on a coordinate plane and draw a smooth curve through the points to represent the parabola y = x^2.

Key Concepts

Ordered Pairs

Ordered pairs are pairs of numbers that represent coordinates on a Cartesian plane, typically written in the form (x, y). In the context of equations, they are used to identify specific solutions where the x-value corresponds to a particular y-value derived from the equation. For example, for the equation y = x^2, the ordered pairs (1, 1), (2, 4), and (3, 9) are solutions since they satisfy the equation.

Graphing Quadratic Functions

Graphing quadratic functions involves plotting the curve of a function defined by a polynomial of degree two, typically in the form y = ax^2 + bx + c. The graph of such functions is a parabola, which opens upwards if 'a' is positive and downwards if 'a' is negative. Understanding the shape and vertex of the parabola is crucial for accurately representing the function visually.

Function Evaluation

Function evaluation is the process of substituting a specific value for the variable in a function to find the corresponding output. For the equation y = x^2, evaluating the function at different x-values allows us to determine the y-values that form ordered pairs. This concept is essential for generating the table of solutions and understanding how changes in x affect y.