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Quadratic equations are a type of polynomial equation where the highest exponent of the variable is 2. The standard form of a quadratic equation is given by \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants. In our exercise, the equation is \( y = x^2 + x - 4 \). This means \( a = 1 \), \( b = 1 \), and \( c = -4 \).
Quadratic equations have a characteristic "U" shape known as a parabola when graphed. The direction of the parabola (upward or downward) depends on the sign of the coefficient \( a \).
If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards. Our equation, \( y = x^2 + x - 4 \), has an upward-opening parabola because \( a = 1 \). Quilting this concept helps to visualize how \( y \) changes as \( x \) changes.

Creating a table of values is an essential step in understanding functions. It provides a visual representation of how the function behaves for different values of \( x \).
To make a table of values for the quadratic equation \( y = x^2 + x - 4 \), you compute \( y \) by substituting each \( x \) value into the equation. For instance, if \( x = -2 \), substitute \( x \) into the equation: \( y = (-2)^2 + (-2) - 4 = 4 - 2 - 4 = -2 \).
Continuing this calculation for each \( x \) value listed will allow you to gather all corresponding \( y \) values. This creates a comprehensive table that maps each input \( x \) to its output \( y \).
The table is a practical tool, helping visualize the relationship between inputs and outputs and gaining a better understanding of how the quadratic function behaves.

In mathematics, the domain of a function refers to all possible input values (\( x \)-values) for which the function is defined, while the range refers to all possible output values (\( y \)-values).
For the quadratic function \( y = x^2 + x - 4 \), the domain is all real numbers because you can substitute any real number for \( x \) without making the equation undefined.
The range, however, is more restricted. Since this function is a quadratic equation with an upward-opening parabola, the range includes all real numbers greater than or equal to its vertex point. The vertex is a critical point where the function changes direction, providing a minimum value of \( y \).
Understanding the domain and range allows you to identify the scope of the function and the extent of its possible outputs.

Function evaluation involves finding the output of a function for specific input values. It is about computing the value of the function given an \( x \) value.
For the function \( y = x^2 + x - 4 \), function evaluation means substituting a specific value of \( x \) into the equation to calculate \( y \).
For example, to find \( y \) when \( x = 2 \), substitute \( x = 2 \) into the function: \( y = (2)^2 + (2) - 4 = 4 + 2 - 4 = 2 \).
Function evaluations help determine specific points on the graph of the function. This process is crucial for building tables of values and plotting graphs accurately, providing a practical understanding of the function's behavior.