Problem 87 - College Algebra

Question

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

Step-by-Step Solution

Answer

The graph of the function \(h(x) = |x + 4|\) is a V-shaped graph similar to the absolute value function \(f(x) = |x|\), but shifted 4 units to the left, intersecting the point (-4, 0).

Step 1: Graph the Absolute Value Function

The absolute value function \(f(x) = |x|\) is a V-shaped graph that intersects the origin (0, 0) and opens upwards. The slope of the line to the left of the y-axis is -1, and the slope of the line to the right of the y-axis is 1.

Step 2: Understand the Transformations

The function \(h(x) = |x + 4|\) can be derived from the function \(f(x) = |x|\) through transformations. In this case, the graph of \(f(x) = |x|\) is shifted 4 units to the left to get the graph of \(h(x) = |x + 4|\). This transformation is a result of replacing \(x\) with \(x + 4\) in the original function.

Step 3: Graph the Transformed Function

To graph \(h(x) = |x + 4|\), start with the graph of \(f(x) = |x|\) and then move every point on the graph 4 units to the left. The x-coordinate of the vertex changes from 0 to -4. Hence, the graph of \(h(x) = |x + 4|\) is a similar V-shaped graph that intersects the point (-4, 0) and opens upwards.

Key Concepts

Function Transformations

Function transformations are changes made to a function's graph by altering its equation. They allow you to modify the position, shape, and orientation of the graph. In the context of absolute value functions, transformations typically involve shifting, stretching, or reflecting the graph.

The most common transformations are:

• Translation: Moving the graph up, down, left, or right.
• Reflection: Flipping the graph over the x-axis or y-axis.
• Stretching/Compressing: Changing the width or height of the graph.

Understanding function transformations is crucial for graphing functions correctly. In this example, the transformation involves a horizontal shift of the absolute value graph.

V-shaped Graph

The absolute value function produces a distinct V-shaped graph. It is defined as the function that takes each input and outputs its non-negative value. Mathematically, this is expressed as:\[ f(x) = |x| \]This basic shape is due to the nature of the absolute value, which transforms negative inputs into positive outputs while maintaining positive inputs unchanged.

Key characteristics of this V-shaped graph:

• Symmetry: The graph is symmetric about the y-axis.
• Vertex: The turning point of the V is called the vertex. For the function \(f(x) = |x|\), the vertex is at the origin (0, 0).
• Linear arms: The arms of the V have linear segments with different slopes: -1 for the left arm and +1 for the right arm.

This shape remains V-shaped regardless of transformations applied to the function.

Shifting Graphs

Shifting a graph means translating it horizontally or vertically without altering its shape. For absolute value functions, this is commonly seen in the form of a horizontal shift.

To shift an absolute value graph horizontally:

• Replace \(x\) with \(x + a\) to move the graph left by \(a\) units.
• Replace \(x\) with \(x - a\) to move the graph right by \(a\) units.

In the function \(h(x) = |x + 4|\), the notation \(x + 4\) indicates a leftward shift of 4 units. This shift means the vertex moves from (0, 0) to (-4, 0), effectively moving every point on the graph 4 units to the left.

Vertical shifts involve replacing \(f(x)\) with \(f(x) + b\) (up by \(b\) units) or \(f(x) - b\) (down by \(b\) units). In our given problem, no vertical shift is applied.

Vertex of a Parabola

In absolute value functions like \(f(x) = |x|\), the vertex is a critical point. While absolute value graphs are not parabolas, the concept of a vertex is similar. The vertex is the point where the graph's direction changes, akin to a turning point.

For the basic function \(f(x) = |x|\), the vertex is at (0, 0).

When transformations are applied, such as in \(h(x) = |x + 4|\), the vertex is shifted to (-4, 0). This shift directly results from the horizontal translation of the graph. Understanding vertex movement helps in sketching transformations accurately, monitoring how changes in the function equation affect the graph's position.

The vertex holds significance in graph analysis as it marks the minimum or maximum point of the function. In the context of absolute value, it represents the lowest point, since V-shaped graphs open upwards.