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Quadratic functions are polynomial functions of the form \( ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the case presented, we made a substitution, turning the original function into a quadratic. This makes it easier to analyze and find its maximum or minimum values. Key characteristics of quadratic functions:

• They graph as parabolas, which can open upwards if \( a > 0 \), or downwards if \( a < 0 \).
• The vertex of the parabola represents the maximum or minimum point.
• The axis of symmetry passes through the vertex, and it can be found using \( x = \frac{-b}{2a} \).

Given the quadratic form obtained through substitution, \( f(t) = -t^2 + 4t + 3 \), we observe a downward-opening parabola. This indicates the vertex will represent a maximum point.

Function substitution is a helpful technique for simplifying complex equations or expressions. By replacing parts of the function with a suitable variable, you can easily understand and solve it. In the given exercise, function substitution simplified \( f(x) = 3 + 4x^2 - x^4 \) to a quadratic form \( f(t) = 3 + 4t - t^2 \) by letting \( t = x^2 \). Why use substitution?

• Simplifies solving: By reducing the complexity of the function, we can tap into well-known methods for solving simpler equations, like quadratics.
• Makes visualization easier: A substitution can help turn a higher-degree polynomial into a standard quadratic, making it easier to graph and analyze.
• Uncovers underlying patterns: Sometimes substitutions reveal symmetries or other features not readily noticeable in the original form.

Thus, substitution becomes a vital tool for delving into the behavior of functions and extracting critical insights such as maximum or minimum values.

The vertex formula is an essential tool for finding the vertex of a quadratic function. For any quadratic function \( f(x) = ax^2 + bx + c \), the vertex \( (h, k) \) can be determined using:\[ h = \frac{-b}{2a} \]Once you find \( h \), substitute back into the function to find \( k = f(h) \), which is either the maximum or minimum value, depending on the parabola's orientation.For our quadratic equation \( f(t) = -t^2 + 4t + 3 \), we calculated:

• \( h = \frac{-4}{2(-1)} = 2 \)
• \( k = f(2) = -4 + 8 + 3 = 7 \)

Since the parabola opens downwards, the vertex (2, 7) indicates a maximum. Consequently, for \( f(x) = 3 + 4x^2 - x^4 \), this maximum value is reached when \( x^2 = 2 \), confirming the maximum value is 7. The vertex formula thus provides a straightforward method to derive crucial function characteristics, especially in cases involving maximum and minimum calculations.