91Ó°ÊÓ

A quadratic function is a type of polynomial function where the highest exponent of the variable is 2. This makes it a degree-2 polynomial. Generally, a quadratic function is expressed in the form \(f(x) = ax^2 + bx + c\). In this equation:

• \(a\), \(b\), and \(c\) are constants.
• \(x\) is the variable.
• \(a\) cannot be zero because the function would then not be quadratic.

Quadratic functions exhibit a "U-shaped" curve called a parabola when graphed. Depending on the sign of the coefficient \(a\), the parabola opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)).
In the exercise given, the function \(f(x) = -x^2 - 4\) is a quadratic function. Here, \(a = -1\), \(b = 0\), and \(c = -4\). The parabola will open downwards because \(a\) is negative. Evaluating the function at specific values gives us the y-values for those x-values on the graph.

The substitution method is a straightforward way to evaluate functions. It involves replacing the variable \(x\) with a specific number to calculate the function's value at that point.
In the given solution, we use the substitution method to find \(f(3)\), \(f(-1)\), and \(f(0)\) for the quadratic function \(f(x) = -x^2 - 4\):

• For \(f(3)\): Substitute \(x = 3\) into the function. This leads to \(f(3) = -(3)^2 - 4 = -9 - 4 = -13\).
• For \(f(-1)\): Substitute \(x = -1\), resulting in \(f(-1) = -(-1)^2 - 4 = -1 - 4 = -5\).
• For \(f(0)\): Substitute \(x = 0\), yielding \(f(0) = -(0)^2 - 4 = -4\).

This method is valuable because it allows us to determine specific points on the graph of the function with ease. Whether you're evaluating linear, quadratic, or other polynomial functions, substitution is a key technique to grasp.

Polynomial functions are expressions that involve sums of powers of variables with coefficients. They have the general form \(anx^n + an-1x^{n-1} + \ldots + a1x + a0\), where each \(a\) is a constant coefficient and \(n\) is a non-negative integer indicating the degree.
Key features of polynomial functions include:

• The degree, which is the highest power of \(x\) with a non-zero coefficient.
• The leading coefficient, which is the coefficient of the highest-degree term.
• A smooth and continuous graph without breaks or holes.

The given function \(f(x) = -x^2 - 4\) is an example of a quadratic, hence a degree-2 polynomial function with a leading coefficient of \(-1\). Polynomial functions can range from simple constant functions (degree 0) to much more complex expressions. Understanding them is crucial, as they form the basis for a considerable portion of algebra and calculus. By analyzing such functions using techniques like substitution, we can predict behavior and solve equations across many fields in mathematics and science.