Problem 33 - College Algebra

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)=2|x+4| $$

Step-by-Step Solution

Answer

To achieve the graph of the function \(h(x) = 2|x + 4|\), start from the basic graph of the absolute value function \(f(x) = |x|\), then shift the entire graph 4 units leftwards and scale vertically by a factor of 2.

Step 1: Graph the Basic Absolute Value Function

First, graph the basic absolute value function, which is \(f(x) = |x|\). This is a V-shaped graph that intersects the x-axis at the origin (0,0). The graph increases to the right and left of the origin, reflecting each other across the y-axis.

Step 2: Apply Horizontal Shift

Once the basic absolute value graph is drawn, apply the horizontal shift. This is done by moving each point on the graph 4 units to the left. This is due to the transformation from |x| to |x+4| as per the given function \(h(x) = 2|x + 4|\). This transformation causes every point on the graph to shift 4 units left along the x-axis.

Step 3: Apply Vertical Scaling

The final transformation to be applied is the vertical scaling. The '2' in the given function shows that every y-coordinate of the graph function should be multiplied by 2. This leads to a vertical stretch, meaning the graph becomes narrower.

Key Concepts

Absolute Value Transformation

When dealing with the absolute value transformation, it's crucial to understand the basic shape of the absolute value function, which is typically a V-shaped graph. Each transformation of this graph can radically change its appearance. In the given example, we start with the simple function f(x)=|x|. By itself, the graph of this function intersects the x-axis at the origin, (0,0), and increases linearly in both directions from that point.

When we introduce transformations like adding a number inside the absolute value, as in |x+4|, or multiplying the outside, as in 2|x|, these simple changes adjust the graph's position and shape. These transformations include horizontal shifts, reflections over the x-axis or y-axis, and vertical stretch or compression. Understanding how these transformations work allows us to modify the basic absolute value graph to match a more complex function, such as h(x)=2|x+4|.

Horizontal Shifts In Graphs

When we talk about horizontal shifts in graphs, we're referring to the movement of a graph left or right along the x-axis. In the solution for the function h(x)=2|x+4|, we observed that the graph shifts each point 4 units to the left. This is identified by the '+4' inside the absolute value notation, resulting in a transformation known as a horizontal shift. Instead of being at the origin, the point where the V-shape starts, or the vertex, is now at (-4,0).

How Do We Spot This Shift?

Generally, a function of the form |x-h| will have a horizontal shift 'h' units to the right if 'h' is positive and 'h' units to the left if 'h' is negative. It's a common mistake to think '+4' moves the graph to the right, but it's quite the opposite. Remember, inside the absolute value, the sign of the number inversely affects the direction of the shift.

Vertical Scaling In Graphs

Vertical scaling is another transformation that changes the shape of a graph. To understand vertical scaling, consider it as stretching or squishing the graph along the y-axis. In the example h(x)=2|x+4|, the coefficient '2' implies that each y-value of the absolute value function is multiplied by 2, resulting in a vertical stretch. The graph becomes narrower as it grows away from the x-axis more steeply.

Determining the Effect of Vertical Scaling

For a function like a|f(x)|, where 'a' is greater than 1, the graph will stretch, making it narrower. If 'a' is between 0 and 1, the graph will compress, making it wider. If 'a' is negative, there will be an additional reflection over the x-axis, which means it will flip upside down while stretching or compressing. Recognizing how these coefficients affect the graph helps us quickly visualize the transformations and accurately graph complex absolute value functions.