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A quadratic function is a polynomial function of the form \[f(x) = ax^2 + bx + c\] where \(a eq 0\), \(b\), and \(c\) are constants. The function is called a 'quadratic' because the highest degree of the variable \(x\) is 2, represented by \(x^2\).

• The shape of the graph of a quadratic function is called a parabola.
• A parabola that opens upwards has a positive coefficient \(a\) and a parabolathat opens downwards has a negative \(a\).

To understand how quadratic functions behave, it’s useful to convert them intodifferent forms, including the Vertex Form, which provides insights into the vertex or thehighest or lowest point of the function. By completing the square, we convert a quadratic function into Vertex Form.

The Vertex Form of a quadratic function is written as: \[f(x) = a(x-h)^2 + k\]where:

• \(a\): Determines the width and the direction in which the parabola opens.
• \(h\): The \(x\)-coordinate of the vertex.
• \(k\): The \(y\)-coordinate of the vertex.

The vertex \((h, k)\) represents the highest or lowest point of the parabola. Converting a quadratic function into Vertex Form involves a process called completing the square. This allows you to easily identify the vertex and understand how the parabola is translated and stretched.

A perfect square trinomial is a specific type of quadratic expression that can be written as the square of a binomial. It has the form: \[a^2 \text{or} (x + b)^2\].
Given a quadratic expression of the form: \(x^2 + bx + c\), we can attempt to turn it into a perfect square trinomial by modifying the constant term\(c\).
The steps to do this are:

• Take half of the coefficient of \(x\) (the linear term), and square it.
• Add and subtract this square value inside the expression.
• Group the perfect square trinomial and complete the square.

For the quadratic function \(f(x) = x^2 - 8x + 18\), we took the coefficient of \(x\) (−8), halved it to get (−4), squared it to get (16), and added and subtracted (16) inside the quadratic function to form the square. Finally, this allowed us to rewrite the function as \((x - 4)^2 + 2\), clearly identifying the perfect square trinomial and writing the quadratic in Vertex Form.