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Understanding what a quadratic function is can aid in solving many mathematical problems. A quadratic function is a type of polynomial function where the highest degree of the variable is 2. This gives the function its characteristic parabolic shape. The general form of a quadratic function is represented as:
\textstyle f(x) = ax^2 + bx + c\textstyle<, where:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

The quadratic function can open upwards or downwards depending on the sign of \(a\). If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards. In our example, we have \(f(x) = -x^2 + 4x + 5\), which means:

• \(a = -1\)
• \(b = 4\)
• \(c = 5\)

This makes the parabola open downwards because \(a\) is negative. This understanding sets the foundation we need to find the vertex.

The vertex of a quadratic function is a crucial point on the graph of the function. It represents the minimum or maximum value the function can take. To determine this vertex, we use the vertex formula. The x-coordinate of the vertex for the quadratic function \(f(x) = ax^2 + bx + c\) is given by the formula:
\textstyle x = -\frac{b}{2a}\textstyle<.
Let's find the x-coordinate for our function where \(a = -1\) and \(b = 4\):
\textstyle x = -\frac{4}{2(-1)} = 2\textstyle<.
Having the x-coordinate of 2, we can now plug it into our function \(f(x) = -x^2 + 4x + 5\) to find the y-coordinate:
\textstyle f(2) = -(2)^2 + 4(2) + 5\textstyle<
= -4 + 8 + 5 = 9.
So, the vertex of the function is at \((2, 9)\). This means that at \(x = 2\), the function reaches its highest point (since the parabola opens downwards) which is \(y = 9\).

The standard form of a quadratic function is vital for understanding its properties and helping to find its vertex efficiently. The standard form is written as \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Knowing this allows us to easily apply the vertex formula and other techniques.

• This form makes it straightforward to identify the coefficients \(a\), \(b\), and \(c\).
• It simplifies the process of finding the vertex and factoring the quadratic if necessary.

For our specific function, \(f(x) = -x^2 + 4x + 5\), we recognized that:
• \(a = -1\)
• \(b = 4\)
• \(c = 5\)
With these values, we can utilize the vertex formula and other quadratic tools to fully analyze the function. Converting quadratic functions to their standard form helps in graphing and understanding their behavior. It also provides clear paths to determine key features such as the vertex, direction of the parabola, and whether the function has a maximum or minimum value.