Problem 37 - Beginning and Intermediate Algebra

Question

Use the transformation techniques to graph each of the following functions. $$g(x)=|x+2|+3$$

Step-by-Step Solution

Answer

To graph the function \(g(x) = |x + 2| + 3\), start with the basic function \(f(x) = |x|\), which is a V-shape with vertex at (0, 0). Apply a horizontal shift of -2 units (move 2 units left) to get the new vertex to (-2, 0), and then apply a vertical shift of +3 units (move 3 units up) to get the final vertex at (-2, 3). The transformed graph will have the same V-shape, with its vertex at (-2, 3) and arms forming a 45-degree angle with the x-axis.

Step 1: Graph the basic function \(f(x) = |x|\)

Firstly, we need to graph the basic function, \(f(x) = |x|\). The graph of this function is a V-shape with its vertex at the origin (0, 0) and its two arms forming a 45-degree angle with the x-axis.

Step 2: Apply horizontal shift

Now we need to apply a horizontal shift to the basic function. The given function is \(g(x) = |x + 2| + 3\). From this, we can identify the horizontal shift as -2 units (since the expression inside the absolute value is x + 2). To apply this shift, we will move the vertex of the basic function 2 units to the left. The new vertex will be at the point (-2, 0).

Step 3: Apply vertical shift

Finally, we need to apply a vertical shift to the function. Looking at the given function again, we can identify the vertical shift as +3 units (from the +3 outside the absolute value). To apply this shift, we will move the vertex of the function from step 2 upwards by 3 units. Our new vertex will be at the point (-2, 3).

Step 4: Graph the transformed function

Now that we have applied the horizontal and vertical shifts, we can graph the transformed function \(g(x) = |x + 2| + 3\). The final graph will have the same V-shape as the basic function, but with its vertex at the point (-2, 3). The two arms of the V will still form a 45-degree angle with the x-axis. In conclusion, the graph of the function \(g(x) = |x + 2| + 3\) can be obtained by applying a horizontal shift of -2 units and a vertical shift of +3 units to the basic function \(f(x) = |x|\).

Key Concepts

Absolute Value Functions

Absolute value functions are a fundamental concept in mathematics, easily recognizable by their unique V-shape graphs. The simplest form of an absolute value function is denoted as \(f(x) = |x|\). Here, the absolute value symbol \(| |\) indicates that the function outputs the non-negative value of its argument \(x\). For example, \(|-3|\) equals 3, and \(|2|\) equals 2. The graph of \(f(x) = |x|\) features its vertex at the origin (0, 0). This point represents the lowest point of the V-shaped graph. The symmetry of this shape is notable, with the arms extending diagonally forming 45-degree angles with the x-axis. Absolute value functions can be modified using transformations, making them versatile for creating various shapes and positions on the coordinate plane.

Horizontal and Vertical Shifts

Transformations like horizontal and vertical shifts alter the position of the graph on the coordinate plane without changing its shape. To understand these transformations, consider the function \(g(x) = |x + 2| + 3\). - **Horizontal Shift:** The expression \(x + 2\) inside the absolute value indicates a horizontal shift. Specifically, it suggests a shift to the left by 2 units. This is because adding a positive number inside the absolute value moves the graph in the negative direction along the x-axis, repositioning the vertex from (0, 0) to (-2, 0). - **Vertical Shift:** Next, consider the +3 outside the absolute value, indicating a vertical shift upwards by 3 units. This moves the vertex from (-2, 0) to (-2, 3), lifting the entire graph higher on the y-axis.Combining these transformations repositions the original graph while maintaining its V-shape, providing flexibility in graph placement and orientation.

Vertex of a Function

The vertex of a function is a critical point that often signifies the maximum or minimum value that a function can attain. In absolute value functions, it represents the turning point of the V-shape. For the basic absolute value function \(f(x) = |x|\), the vertex is initially located at the origin (0, 0). As transformations are applied, this point shifts. In our example with \(g(x) = |x + 2| + 3\), transformations move the vertex to (-2, 3).Identifying and understanding the vertex's location is crucial. It helps in graphing and provides insights into function behaviour. It's often the first step when assessing function transformations, and serves as a reference point for further graph adjustments.

Graphing Techniques

Graphing an absolute value function after transformations is systematic and relies on understanding the effects of each transformation. Here’s a basic method:- **Start with the Basic Graph:** Begin by sketching the graph of \(f(x) = |x|\). This forms the template for further alterations.- **Apply Horizontal Shift:** Identify any horizontal changes. For \(g(x) = |x+2| + 3\), move the entire graph 2 units to the left, shifting the vertex to (-2, 0).- **Apply Vertical Shift:** Next, apply vertical adjustments. In the example, shift the graph 3 units up, repositioning the vertex to (-2, 3).- **Confirm Shape and Angles:** Ensure that the V-shape and 45-degree arm angles are maintained during shifts. This check guarantees graph accuracy.Graphing techniques make it possible to visualize the transformations in absolute value functions with precision, guiding us to depict complex functions effectively through basic and modified graphs.