Problem 30 - College Algebra

Question

For the following exercises, graph the given functions by hand. $$ f(x)=-|x+4|-3 $$

Step-by-Step Solution

Answer

The graph of \\( f(x) = -|x+4|-3 \\) is a V-shape with vertex \\( (-4, -3) \\) opening downwards.

Step 1: Identify and Understand the Base Function

The base function here is the absolute value function, which is expressed as \( |x| \). The absolute value function produces a V-shaped graph with the vertex at the origin.

Step 2: Apply Transformations to the Base Function

The given function is \( f(x) = -|x+4|-3 \). There are two transformations applied to the base function:1. Horizontal Translation: \( |x+4| \) translates the graph 4 units to the left, shifting the vertex of the V.2. Vertical Translation and Reflection: The negative sign in \(-|x+4|\) reflects the graph across the x-axis, and subtracting 3 shifts the graph 3 units downward.

Step 3: Plot the Transformed Vertex

With these transformations, the new vertex of the graph is located at \((-4, -3)\). Plot this point on a coordinate plane as it represents the lowest point in this transformed V-shaped graph because of the reflection.

Step 4: Draw the V-Shaped Graph

From the vertex \((-4, -3)\), draw lines extending upwards and outwards in both directions to form the V shape. Since the absolute value function usually has a slope of 1, the lines will rise 1 unit for every 1 unit you move away from the vertex horizontally. However, due to the reflection, they will actually move downwards as they move away from the vertex horizontally.

Step 5: Verify the Graph

Check a few additional points to ensure accuracy, such as calculating \( f(x) \) for \( x = -6 \) and \( x = -2 \) to ensure the plots fall on the lines you've drawn. Confirm that for \( x = -6 \), \( f(-6) = -5 \), and for \( x = -2 \), \( f(-2) = -5 \). This ensures the lines drawn follow the slope and produce the correct transformed graph.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics. Represented as \( |x| \), this function creates a distinct V-shaped curve on a graph. The key characteristic of the absolute value function is that it always outputs non-negative values, regardless of whether the input \( x \) is positive or negative.

• For positive or zero values of \( x \), \( |x| = x \).
• For negative values of \( x \), \( |x| = -x \), resulting in positive outputs.

This basic property makes the graph symmetrical with respect to the y-axis, with its vertex positioned at the origin \((0, 0)\). Understanding this basic nature is crucial when further transformations are applied. The absolute value function is notably used in diverse fields, helping represent problems involving distances, among others.

Transformations of Functions

Transformations modify the appearance of the base function's graph without altering its fundamental nature. These changes can include shifting, stretching, shrinking, and reflecting the graph. For any function \( g(x) \), a transformation can be expressed in terms of a function like \( f(x) \):

• Shifting involves moving the graph horizontally or vertically without rotating it.
• Reflecting modifies the orientation by flipping the graph over a given axis.
• Stretching or shrinking involves changing the size of the graph along the x or y-axis, modifying its steepness or width.

In the case of \( f(x) = -|x+4|-3 \), multiple transformations occur:- The reflection across the x-axis, denoted by the negative outside the absolute value, inverts the V-shape. - Horizontal translations (shifts along the x-axis) and vertical translations (shifts along the y-axis) further adjust the position on the plane.These combined transformations result in a new vertex at \((-4, -3)\). Recognizing these alterations, especially in tests, allows one to quickly sketch or interpret the modified functions.

Vertical and Horizontal Shifts

Vertical and horizontal shifts are specific types of transformations that move a graph along the y-axis or x-axis.**Horizontal Shifts:**A horizontal shift moves a graph left or right. To achieve this, you alter the input value \( x \) within the function. For example, transforming \( |x| \) to \( |x+4| \) shifts the graph 4 units to the left. Here, the entire graph, including the vertex, moves in the direction specified by the expression inside the absolute value function.**Vertical Shifts:**Vertical shifts occur when the output of the function is altered, effectively moving the graph up or down the y-axis. In the function \( f(x) = -|x+4|-3 \), subtracting 3 translates the graph downwards by 3 units.Both shifts are often easy to apply and visualize:- Horizontal shifts are typically indicated by additions or subtractions inside the function.- Vertical shifts result from changes outside the function, modifying its output directly.Understanding these shifts is essential when graphing complex functions, making it simpler to predict and draw the graph location effectively.