91Ó°ÊÓ

Quadratic functions are a foundational concept in mathematics, particularly in algebra. These functions are characterized by their equation format, typically written as \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, with \( a eq 0 \).
In our exercise, both \( f(x) = x^2 \) and \( g(x) = x^2 - 2 \) are quadratic functions, represented in the standard form of \( ax^2 + bx + c \). Specifically:

• \( f(x) = x^2 \) corresponds to \( a = 1, b = 0, \) and \( c = 0 \).
• \( g(x) = x^2 - 2 \) has \( a = 1, b = 0, \) and \( c = -2 \).

These functions form parabolas, which are symmetrical, U-shaped curves. Understanding the parameters of a quadratic function helps in predicting and analyzing the position and direction of the parabola on the graph.

The graph of any quadratic function is a parabola. Recognizing and interpreting parabolas is essential when studying quadratic functions. In the simplest terms, a parabola can open upwards or downwards, and its central point is known as the vertex.
For the function \( f(x) = x^2 \), the parabola opens upwards with its vertex at the origin \((0,0)\). This is a standard representation where the parabola is centered, and is symmetrical along the y-axis.
The other function, \( g(x) = x^2 - 2 \), also forms a parabola that has the same upward-facing shape. However, its vertex is at \((0, -2)\), shifted vertically from the origin. Despite their different positions, both parabolas share the axis of symmetry. This means they remain identical in form, only varying in position on the graph plane.

A vertical shift is a type of transformation applied to functions, which moves a graph up or down on the coordinate plane without altering its shape.
In the case of the transformation from \( f(x) = x^2 \) to \( g(x) = x^2 - 2 \), the graph experiences a vertical shift of 2 units downward. This type of transformation is identified by the constant term—called \( c \)—in the quadratic equation \( ax^2 + bx + c \).

• When \( c \) is positive, the graph shifts upward.
• When \( c \) is negative, the graph shifts downward.

For \( g(x) = x^2 - 2 \), the \(-2\) indicates a downward shift of 2 units from the standard position of \( f(x) \). As a result, the vertex moves from \( (0,0) \) to \( (0,-2) \), effectively altering the graph's vertical placement without affecting the parabola's width or direction.