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A quadratic function is a type of polynomial function that is characterized by its standard form: \( f(x) = ax^2 + bx + c \). In this form, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \).
In the problem we are considering, the function given is \( f(x) = -2(x+1)^2 - 4 \). This is a transformed version of the standard quadratic function. Here, the expression \((x + 1)^2\) indicates a shifted form of the basic quadratic \(x^2\). When expanded, it produces \(x^2\) terms that create the parabola shape, but the form \( (x+1) \) shows the entire graph is shifted to the left by 1 unit.
Moreover, the coefficient \(-2\) in front of \((x+1)^2\) indicates a vertical stretching and reflection of the basic parabola. This means the parabola opens downward, making it look like a frown, and the vertical stretch makes the graph narrower than the typical \(x^2\) graph. The constant \(-4\) shifts the entire graph downward by 4 units.

The substitution method is a straightforward approach used to evaluate functions at specific points. This technique is particularly helpful in mathematics for functions like quadratics.
To evaluate a function using substitution:

• Take the given function, for example, \( f(x) = -2(x+1)^2 - 4 \).
• Replace \( x \) with the specific number you are evaluating. In this exercise, for \( f(3) \), substitute \( 3 \) for \( x \).
• Make sure to substitute correctly and adhere strictly to mathematical operations.

Once substitution is complete, the next steps involve handling the arithmetic, which leads us to the next important concept, algebraic manipulation.

Algebraic manipulation involves rearranging mathematical expressions and equations to simplify or solve them. When evaluating functions, especially quadratics, this skill is essential to arrive at a clear answer.
After substituting, you'll generally perform several steps:

• Simplify expressions inside parentheses first, such as \((3+1)\).
• Handle exponents next, like squaring \((4)^2\) to get 16.
• Apply multiplication then, for instance, \(-2 \times 16\), giving \(-32\).
• Finally, add or subtract any constants as the last step, like \(-32 - 4 = -36\).

These steps require paying careful attention to order of operations (PEMDAS/BODMAS). Practicing algebraic manipulation helps you accurately evaluate functions and strengthens your overall mathematical foundation.