Problem 48 - Algebra 2

Question

Graph each pair of functions on the same coordinate plane. Describe the translation that takes the first function to the second function. \(y=|x+1|, y=|x-5|\)

Step-by-Step Solution

Answer

The translation that takes the function \( y=|x+1| \) to \( y=|x-5| \) is a horizontal shift 6 units to the right.

Step 1: Graph the first function

Plot the function, \(y=|x+1|\), on the coordinate plane. This is an absolute value function, which looks like a 'V' shape. The vertex of the 'V' is translated 1 unit left from the origin since it is \(x+1\) inside the absolute value.

Step 2: Graph the second function

Plot the function, \(y=|x-5|\), on the same coordinate plane. This function also has the shape of a 'V'. The vertex of the 'V' is translated 5 units right from the origin because of the \(x-5\) inside the absolute value.

Step 3: Identify the translation

Observe the difference in the positions of the vertices of the two functions. The vertex of the first function is at \((-1, 0)\), and the vertex of the second is at \((5, 0)\). To translate the first function to the second, we need to move it 6 units to the right.

Key Concepts

Absolute Value Graphs

Understanding the graph of an absolute value function is essential when working with various mathematical concepts. An absolute value graph typically has a 'V' shape, with the point at which the lines meet, known as the vertex, acting as a pivot. For the standard function, represented as \(y = |x|\), the vertex is at the origin (0,0).

When numbers are added or subtracted within the absolute value, such as \(y = |x + 1|\) or \(y = |x - 5|\), the graph shifts, or 'translates', along the horizontal axis. If the number is positive, the graph moves to the left; if it's negative, the graph shifts to the right. This attribute helps us quickly identify how the graph of a standard absolute value function has been altered and understand the resultant graph's positioning on the coordinate plane.

Graphing Transformations

Graphing transformations involve moving or changing the size, shape, or orientation of a graph without altering its basic structure. Transformations can include translations (slides), reflections (flips), rotations (turns), and dilations (stretches or shrinks).

In the context of absolute value graphs, translations are the most relevant. To execute a translation, every point of the graph is moved the same distance in the same direction. The direction of the shift is determined by sign and magnitude of numbers added to or subtracted from \(x\) or \(y\) in the equation. Graphing these transformations correctly ensures that the underlying relationship between variables is accurately represented in a visual format.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is determined by its position along the x (horizontal) and y (vertical) axes. The intersection point of these axes is called the origin and is designated as (0,0). Coordinates are written as ordered pairs (x, y), representing the point's horizontal and vertical displacement from the origin.

The concept of the coordinate plane is fundamental in graphing as it provides a framework to display algebraic equations visually. Understanding how to navigate this plane allows us to interpret and manipulate graphs of functions, such as translating the absolute value function from one position to another.

Function Translation

Function translation is an operation that shifts a graph horizontally, vertically, or both, without changing its shape or orientation. When we describe a function translation, we outline how far and in which direction the graph has moved.

For example, to translate the function \(y = |x+1|\) to \(y = |x-5|\), we notice that the vertex moved from point (-1, 0) to (5, 0), indicating a horizontal shift of 6 units to the right. Identifying translations between graphs of functions helps us understand the relationship between these functions and their algebraic representations, as well as providing a clear method to describe how one function morphs into another.