Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Describe the sequence of transformations from \(f(x)=|x|\) to \(g .\) Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=|x+1|-3\)

Step-by-Step Solution

Answer

The function \(g(x)=|x+1|-3\) is a transformation of the base function \(f(x)=|x|\), where there is a shift 1 unit to the left and 3 units down. After sketching this on a graph, you should have an absolute value function graph opening upwards with a vertex at (-1, -3).

Step 1: Identify the Transformations

The given function \(g(x)=|x+1|-3\) is a transformation of \(f(x) = |x|\). Let's break it down: \n1. The +1 inside the absolute value causes a shift of 1 unit to the left. 2. The -3 outside the absolute value causes a downward shift of 3 units.

Step 2: Sketch the Transformed Graph

Start with the graph of \(f(x) = |x|\). The point where the graph turns, called the vertex, begins at the origin, (0,0). 1. Move the vertex 1 unit to the left to account for the +1 inside the absolute value. Now the vertex sits at (-1, 0).2. Then, move the vertex down by 3 units, to account for the -3 outside of the absolute value. Now the vertex sits at (-1, -3). The function \(g(x)\) now has the graph of an absolute value function, opening upwards with its vertex at (-1,-3).

Step 3: Verify the Graph

Use a graphing utility or calculator to graph the function \(g(x) = |x+1|-3\). Comparing the manual sketch of the graph with the one generated by the utility should confirm that both graphs match up.

Key Concepts

Absolute Value Transformations

Absolute value transformations involve changing the basic structure of an absolute value function. The function \(f(x) = |x|\) is the most basic absolute value function and has a V-shaped graph that opens upwards. When you transform \(f(x)\), you're essentially altering its position or shape on the coordinate plane. These transformations can include:

• Translations: Moving the graph vertically or horizontally without changing its shape.
• Reflections: Flipping the graph over a line, such as the x-axis or y-axis.
• Stretches/Compressions: Making the graph narrower or wider.

For \(g(x) = |x+1|-3\), the transformations include a shift to the left and downward. Understanding these changes helps in predicting the new position of the graph on the coordinate plane.

Graphing Absolute Value Functions

Graphing absolute value functions involves plotting a V-shaped graph based on transformations to the basic \(y = |x|\) graph. The graph of \(|x|\) has a vertex at the origin, (0,0), unless shifted. To graph \(g(x) = |x+1|-3\), start with the graph of \(f(x) = |x|\):

• The vertex is initially at (0,0).
• Shift this vertex according to the transformations identified. Here, move 1 unit left to (-1,0).
• Then, shift the vertex down to (-1,-3).

This results in the new function \(g(x)\), where the V-shaped graph opens upwards from the vertex at (-1,-3). The slopes of the arms of the V remain the same as those of the original \(|x|\), typically a slope of 1 or -1 from the vertex.

Vertical and Horizontal Shifts

Vertical and horizontal shifts are types of translations that relocate the entire graph of a function without altering its shape. For vertical shifts:

• Upward shift: Occurs by adding a constant outside the function, moving it up.
• Downward shift: Occurs by subtracting a constant outside the function, moving it down.

For \(g(x) = |x+1|-3\), the \(-3\) causes a downward shift by 3 units.For horizontal shifts:

• Right shift: Achieved by subtracting a constant inside the function, moving it right.
• Left shift: Achieved by adding a constant inside the function, moving it left.

In our function, the \(+1\) inside the absolute value leads to a left shift by 1 unit. These shifts ultimately let us reposition the graph, helping us accurately depict \(g(x) = |x+1|-3\) on the coordinate plane.