Problem 91 - 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

In conclusion, the given function \(h(x) = 2|x+4|\) can be graphed by starting with the graph of the absolute value function \(f(x) = |x|\), shifting it 4 units to the left, and then stretching it vertically by a factor of 2.

Step 1: Graph the Absolute Value Function

Start with the graph of the absolute value function, \(f(x) = |x|\). This is a V-shaped graph that intersects the y-axis at (0,0) and opens upwards. The x-axis is the line of symmetry.

Step 2: Perform Horizontal Shift

The given function is \(h(x) = 2|x+4|\), as compared to the base function, it has a '+4' inside the absolute value. This represents a horizontal shift of 4 units to the left. Therefore, each point on the graph of \(y = |x|\) should be moved 4 units to the left to get the graph of \(y = |x+4|\).

Step 3: Perform Vertical Scaling

The '2' in front of the absolute value in the function \(h(x) = 2|x+4|\) represents a vertical stretching by a factor of 2. So, any point (x, y) on the graph of \(y = |x+4|\) will be transformed to the point (x, 2y) on the graph of the given function.

Key Concepts

Understanding the Absolute Value Function

The absolute value function, represented mathematically as \( f(x) = |x| \), is a crucial concept in the study of functions and transformations. This function forms a distinct, V-shaped graph, which is symmetric around the y-axis and originates at the point (0,0). The absolute value is known for its ability to convert negative inputs into positive outputs, while leaving positive inputs unchanged. This means that for any real number \( x \), the output is always zero or positive, creating a graph that mirrors itself on either side of the y-axis.
The basic properties of the absolute value function include:

• The vertex is at the origin (0,0).
• The graph opens upwards.
• The line of symmetry is the y-axis.

These characteristics make the absolute value function a powerful tool for illustrating how other functions behave when adjusted through transformations.

The Concept of Horizontal Shifts

Horizontal shifts are a type of transformation that affect the x-values of a function's graph. In the case of the absolute value function, a shift is applied by adding or subtracting a constant within the absolute value symbol. For the function \( h(x) = 2|x+4| \), the term '+4' inside the absolute value results in a horizontal shift to the left by 4 units.
This can be understood by considering how each x-coordinate on the graph of \( |x| \) is decreased by 4, effectively moving every point on the curve to the left. This technique is crucial for analyzing the behavior of functions and understanding how basic graphs can be manipulated.
Remember:

• Adding a positive value inside the absolute value function shifts the graph to the left.
• Subtracting a positive value shifts it to the right.

This situation often feels counterintuitive, but keeping these rules in mind will clarify how graphs shift horizontally.

Vertical Scaling and Its Impact

Vertical scaling modifies the steepness and height of a graph. For the function \( h(x) = 2|x+4| \), the coefficient ‘2’ in front of the absolute value dictates this vertical scaling. It stretches the graph vertically by a factor of 2. In simple terms, for every point on the initial graph of \( y = |x+4| \), the new y-coordinate is doubled.
When dealing with vertical scaling:

• A factor greater than 1 results in the graph stretching away from the x-axis, making it appear steeper.
• A factor between 0 and 1 compresses the graph towards the x-axis, making it appear flatter.

Vertical scaling is an essential transformation that allows us to comprehend how the amplitude or steepness of graphs can be altered without affecting the overall shape or direction. This transformation, combined with horizontal shifts, can significantly change the appearance and function of a graph while preserving its fundamental properties.