Problem 83 - College Algebra

Question

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)-|x+4| $$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = |x + 4|\) will be a V-shape identical to the base function \(f(x) = |x|\), but shifted 4 units to the left. So, the vertex of the V is now at (-4, 0) instead of at the origin (0,0).

Step 1: Understand the Base Graph

First, let's understand the absolute value base function \(f(x) = |x|\). The graph of \(|x|\) is a V-shape which intersects the origin (0,0), and for any \(x > 0\), \(y = x\) and for \(x < 0\), \(y = -x\).

Step 2: Understand the Transformation

The transformed function \(g(x) = |x + 4|\) includes a horizontal shift of the base function. The graph of \(|x + 4|\) will shift 4 units to the left from the original graph.

Step 3: Graph the Transformed Function

To graph the function \(g(x) = |x + 4|\), follow the same shape as the base function \(f(x) = |x|\), but shift each point 4 units to the left. It has the same shape as the base function; the vertex of the V is now at (-4, 0). For \(x > -4\), \(y = x + 4\) and for \(x < -4\) \(y = -x - 4\).

Key Concepts

Understanding Absolute Value Transformation

When we talk about absolute value transformations, we're discussing changes to the basic absolute value function, usually represented as f(x) = |x|. The graph of this function has a distinct 'V' shape with its lowest point at the origin (0,0). Transforming this function can involve shifting it horizontally or vertically, reflecting it across an axis, or stretching it.

Let's consider the function g(x) = |x + 4|. The '+4' inside the absolute value symbol indicates a horizontal shift. Unlike what we might initially think, the function moves 4 units to the left rather than to the right. This is because we're adding the number before taking the absolute value, which affects the direction of the shift opposite to the sign of the number.

An absolute value transformation respects the basic 'V' shape of the function. It doesn't flatten or stretch the 'V'; it just moves it around the graph. As with any transformation, it's crucial to maintain the symmetry of the graph while moving it.

Grasping the V-shape Graph

The V-shape graph is the most defining characteristic of the absolute value function. Envision opening a book to where it naturally rests - this represents the V-shape created by the function f(x) = |x|. The point at which the book's spine would be is called the vertex, the lowest point in this function, and for the absolute value, it starts at the origin.

Understanding Symmetry

One side of the 'V' is the mirror image of the other. For the absolute value function, the line of symmetry runs vertically through the vertex. If you were to fold the graph along this line, one side would perfectly match up with the other.

Piecewise Description

The absolute value function can be broken down into two lines - one with a positive slope for x-values greater than zero, and the other with a negative slope for x-values less than zero. This makes it easy to plot the graph and understand its behavior.

Analyzing Horizontal Shift

A horizontal shift moves the graph of a function left or right across the x-axis. In the context of the absolute value function, a horizontal shift can significantly alter where the V-shape appears on the graph.

Consider the function g(x) = |x + 4| from our original problem. In this case, the horizontal shift is by 4 units to the left because of the '+4' inside the absolute value. It's important not to confuse this with moving to the right; the positive inside the absolute value notation always moves the graph to the left, opposite its sign.

How to Visualize the Shift

Imagine sliding the entire graph of the base function f(x) = |x| to the left without altering its shape. Each point on the graph moves the same distance in the same direction - this is the horizontal shift in action. By understanding this concept, graphing transformed absolute value functions will become a more intuitive process.