91Ó°ÊÓ

The standard quadratic form refers to the representation of a quadratic function in the equation format \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are coefficients, and \( a \) is not equal to zero. The factors \( a, b, \) and \( c \) determine the shape and position of the parabola on the Cartesian plane.

In the provided exercise, \( h(x)=(x+4)^2 \) needed to be expanded to find the standard quadratic form, which results in \( h(x)=x^2+8x+16 \). This process showcases how we go from a compact, binomial expression to a standard quadratic form. Understanding how the equation transitions from one form to another helps students recognize the characteristics of the graph, such as direction of opening and width. In this case, since \( a=1 \), the parabola opens upwards and has a standard width.

The vertex of a quadratic function is a crucial point where the function reaches its maximum or minimum value. In a standard quadratic form like \( ax^2+bx+c \), the vertex \( (h, k) \) can be found using the formula \( h = -\frac{b}{2a} \) and \( k = f(h) \), or \( k = ax^2 + bx + c \) with \( x=h \).

The vertex is not only an essential aspect of the graph's shape but also serves as a reference for symmetry. In our exercise, using the formula \( h = -\frac{b}{2a} \) with \( b = 8 \) and \( a = 1 \) reveals that the vertex of the parabola described by \( h(x)=(x+4)^2 \) is at \( (-4, 0) \) since the function's value at \( x=-4 \) is 0. Students should note that the vertex is a central concept in describing and understanding the properties of quadratic functions.

Expanding binomials is a fundamental skill in algebra that involves rewriting a binomial expression, such as \( (x+4)^2 \) from our exercise, into its expanded form using algebraic operations. The process follows the distributive property, often applied by using the FOIL method (First, Outer, Inner, Last) for multiplying two binomials.

In the context of quadratic functions, expanding a binomial squared, like in our exercise, reveals the coefficients that define the standard quadratic form \( ax^2+bx+c \). It translates the compact form into an expanded form that clearly displays all the terms affecting the parabola's shape and position. For example, expanding \( (x+4)^2 \) gives \( x^2+8x+16 \) after applying the distributive property, showing the individual terms that contribute to the standard form.