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Understanding the domain and range of a quadratic function like the one in our example is crucial. The **domain** of a function is all the possible values that the variable x can take. For the given quadratic equation, \( y = x^2 - 1 \), there are no restrictions on the values x can have. This means the domain is all real numbers, represented in interval notation as \( (-\infty, \infty) \).

The **range** of a function refers to all possible output values (y-values). For the equation \( y = x^2 - 1 \), the smallest value of y occurs when x is zero. If we substitute x = 0 into the equation, we get \( y = 0^2 - 1 = -1 \). As x moves away from zero, the y-value increases. Hence, the range is all real numbers starting from -1 to infinity, written as \( [-1, \infty) \).

Determining the domain and range helps us understand the extent of the function's graph along the x and y axes.

Graphing quadratic equations lets us visually interpret the relationship between x and y. For the equation \( y = x^2 - 1 \), graphing involves plotting ordered pairs and drawing the shape they form.

To start, choose a set of x-values. Common choices include integers like -3, -2, -1, 0, 1, 2, and 3.

Next, calculate the corresponding y-values. For example, if x = 2, then y = 2^2 - 1 = 3. The pairs (x, y) obtained are: \( (-3, 8), (-2, 3), (-1, 0), (0, -1), (1, 0), (2, 3), (3, 8) \).

Plot these points on graph paper or a coordinate plane. After marking the points, connect them to reveal the curve of the equation. Here, the graph is a parabola that opens upward with the vertex at (0, -1).

Graphing makes it easier to understand and analyze the function's behavior.

Ordered pairs are essential in graphing equations. An **ordered pair** is a pair of numbers (x, y) that represent coordinates on the graph.

Finding ordered pairs involves choosing values for x and computing the corresponding y-values using the given equation. For instance, using the equation \( y = x^2 - 1 \), if we pick x = -2, substituting it in gives us \( y = (-2)^2 - 1 = 3 \). Thus, (-2, 3) is an ordered pair.

Creating a list of these pairs helps us draw the curve of the function. For our equation, the list includes: \( (-3, 8), (-2, 3), (-1, 0), (0, -1), (1, 0), (2, 3), (3, 8) \).

Ordered pairs visualize how changes in x affect y and depict the pattern of increase or decrease in values. This understanding is foundational for comprehending more complex functions.