Problem 88 - Calculus Volume 1

Question

Use the graph of \(f(x)=x^{2}\) to graph each transformed function \(g\). \(g(x)=x^{2}-1\)

Step-by-Step Solution

Answer

Graph \( f(x) = x^2 \) shifted down by 1 unit.

Step 1: Identify Parent Function

The parent function given is \( f(x) = x^2 \), which is a basic quadratic function that graphs as a parabola opening upwards with the vertex at the origin (0,0).

Step 2: Identify Transformation

The function \( g(x) = x^2 - 1 \) represents a vertical shift of the parent function \( f(x) \). The \(-1\) indicates that the graph of \( f(x) \) will be shifted downward by 1 unit. This doesn't affect the shape of the graph, only its vertical position.

Step 3: Graph the Transformed Function

To graph \( g(x) = x^2 - 1 \), take the graph of \( f(x) = x^2 \) and move each point on this graph down by 1 unit. The vertex of the parabola moves from (0,0) to (0,-1), and other points follow this downward shift accordingly.

Key Concepts

Understanding the Parabola

A parabola is a smooth, symmetrical curve shaped like the letter U. This curve is the graphical representation of a quadratic function, typically taking the form of \( f(x) = ax^2 + bx + c \). When we deal with a parabola, we're looking at a quadratic function where the variable \( x \) is squared, producing this unique, upward or downward opening shape.
The most fundamental form of a parabola occurs in the parent function \( f(x) = x^2 \). This simple function forms a standard parabola when graphed. It is set perfectly symmetrical along the \( y \)-axis and has its vertex at the origin, which is the point (0,0).

• Its shape and direction are determined by the coefficient of \( x^2 \).
• A positive coefficient means the parabola opens upwards.
• A negative coefficient would make it open downwards.

Understanding the basic structure and features of a parabola like symmetry, vertex, and direction is key when learning about quadratic functions and their transformations.

Exploring Graph Transformations

Graph transformations allow us to change the appearance and position of graphs without altering their basic shape. They help describe how the graph of a function is shifted, stretched, or flipped. With quadratic functions, these transformations are particularly interesting because they affect the parabola in predictable ways.
In the case of the transformation \( g(x) = x^2 - 1 \), we're focusing on a vertical shift. A vertical shift involves moving the entire graph up or down along the \( y \)-axis.

• Adding to \( x^2 \) moves the parabola up.
• Subtracting from \( x^2 \) shifts it down.

In \( g(x) = x^2 - 1 \), the transformation involves taking each point on the parent parabola \( f(x) = x^2 \) and moving it 1 unit down. This means that while the shape of the graph remains the same—a U-shaped parabola—the vertex moves from the origin (0,0) to (0,-1).
Other types of transformations include horizontal shifts, reflections, and stretches or compressions, each altering the graph's position, orientation, or size.

Recognizing the Parent Function

The parent function is the simplest form of a function family. For quadratic functions, the parent function is \( f(x) = x^2 \). Knowing this function is essential because it's the starting point from which other quadratic functions—its transformations—are derived.

The parent function \( f(x) = x^2 \) features a parabola:

• Symmetrical about the \( y \)-axis.
• Vertex located at (0,0).
• Opening upwards because the coefficient of \( x^2 \) is positive.

When graphing transformations like \( g(x) = x^2 - 1 \), our understanding of the parent function allows us to easily identify what changes have occurred. The parent function helps in understanding the effect of various transformations such as shifts, reflections, or stretches.
By mastering the parent quadratic function, graphing any transformed version becomes an intuitive process because you can simply visualize changes from this basic graph.