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A quadratic function is a type of polynomial function where the highest degree of the variable is squared, meaning it has a degree of 2. The general form of a quadratic function is given by: \[ f(x) = ax^2 + bx + c \]Where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the given exercise, our function is \( f(x) = x^2 - 6 \). Here, \( a = 1 \), \( b = 0 \), and \( c = -6 \). This makes it one of the simplest forms of quadratic functions where the linear term, represented by \( bx \), is not present. Quadratic functions have a parabolic shape when graphed, meaning they either open upwards or downwards. In our case, since the coefficient \( a \) is positive, the parabola opens upwards. The vertex of this parabola, representing its minimum point, would be at the line of symmetry, which is \( x = 0 \), resulting in the output value \( f(0) = -6 \). This model represents real-world scenarios such as projectile motion or area calculations.

Function evaluation is the process of finding the output of a function for specific input values. In simpler terms, it is asking the question "What value do you get when you put a specific number into the function?". Given the function \( f(x) = x^2 - 6 \), you evaluate the function at certain values by substituting \( x \) with those specific values. For example, to evaluate \( f(3) \):- Substitute \( x \) with 3: \[ f(3) = 3^2 - 6 \]- Calculate: \[ f(3) = 9 - 6 = 3 \]So, \( f(3) \) equals 3. Repeating this with any number gives the corresponding output of the function for that input. This is essential in understanding how functions respond to different inputs and finding patterns or behaviors in mathematical and real-world contexts.

The substitution method is a straightforward and effective way to evaluate a function. This technique involves substituting a given value into the place of the variable in the function and then performing the arithmetic operations to find the output. For the function \( f(x) = x^2 - 6 \), substitution is how we determine values like \( f(0) \) or \( f\left( \frac{1}{2} \right) \). For example:- To find \( f(-3) \), substitute \( x \) with -3: \[ f(-3) = (-3)^2 - 6 = 9 - 6 = 3 \]Each step involves replacing \( x \) with the specific number and then calculating the result according to the operations defined by the function. The substitution method is a fundamental skill in mathematics that allows for flexibility and understanding in evaluating different kinds of functions.