Problem 85 - Intermediate Algebra

Question

How are the graphs of \(f(x)=x^{2}\) and \(g(x)=x^{2}-2\) related?

Step-by-Step Solution

Answer

The graph of \(g(x)\) is the graph of \(f(x)\) shifted 2 units downwards.

Step 1: Understand the Functions

The function \(f(x) = x^2\) is a basic quadratic function, representing a parabola that opens upwards and has its vertex at the origin \((0,0)\). The function \(g(x) = x^2 - 2\) is another quadratic function, similar in shape to \(f(x)\) but with a vertical transformation.

Step 2: Analyze the Transformation of \(g(x)\)

The function \(g(x) = x^2 - 2\) can be seen as \(f(x) = x^2\) subtracted by 2. This means that for every value of \(x\), the output of \(g(x)\) is 2 units less than that of \(f(x)\). This indicates that \(g(x)\) is a vertical shift of \(f(x)\) downwards by 2 units.

Step 3: Describe the Graphs' Relationship

Both functions have the same shape—a parabola opening upwards—but the graph of \(g(x)\) is translated 2 units downwards compared to the graph of \(f(x)\). Thus, the vertex of \(f(x)\) is \((0,0)\) while the vertex of \(g(x)\) is \((0,-2)\).

Key Concepts

Quadratic Functions

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.

Parabolas

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.

Vertical Shift

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.