Problem 69 - Precalculus : Building Concepts and Connections

Question

Use a graphing utility to solve the problem. Graph \(f(x)=|x+3.5|\) and \(g(x)=|x|+3.5 .\) Describe each graph in terms of transformations of the graph of \(h(x)=|x|\).

Step-by-Step Solution

Answer

The function \(f(x) = |x+3.5|\) is the graph of \(h(x)=|x|\) shifted 3.5 units to the left. The function \(g(x)=|x|+3.5\) is the graph of \(h(x)=|x|\) shifted 3.5 units upwards.

Step 1: Graph the Functions

Begin by using your graphing utility to create a graph for each function: \(f(x) = |x+3.5|\) and \(g(x) = |x| + 3.5\). Also, graph the base function \(h(x) = |x|\) for reference.

Step 2: Analyze the Graph of \(f(x)\)

The graph of \(f(x)=|x+3.5|\) is a transformation of the graph of \(h(x)=|x|\). The expression inside the absolute value, \(x+3.5\), represents a shift or translation of the graph. Since it's \(x+3.5\), it's essentially \(x-(-3.5)\), which means it is a horizontal shift to the left by 3.5 units.

Step 3: Analyze the Graph of \(g(x)\)

For the function \(g(x)=|x|+3.5\), the \(+3.5\) that's outside the absolute value function represents a vertical shift. It's a shift upwards by 3.5 units.

Step 4: Comparing the Transformations

Comparing all the graphs, we can see that \(f(x) = |x+3.5|\) corresponds to a horizontal shift of the basic \(h(x) = |x|\) graph to the left by 3.5 units, while \(g(x) = |x|+3.5\) corresponds to a vertical shift upwards by 3.5 units. Thus, you can describe each graph in terms of transformations of the base function \(h(x) = |x|\).

Key Concepts

Graph Transformations

Graph transformations help us understand how a base graph is altered by changes to its equation. When dealing with absolute value functions, these transformations include shifting, reflecting, and stretching or compressing. For the functions in our problem, we examine two types of shifts. By transforming the base graph \( h(x) = |x| \), we gain insights into how these two functions, \( f(x) = |x + 3.5| \) and \( g(x) = |x| + 3.5 \), have been modified from \( h(x) \). These transformations make it easier to predict the graphs' new positions without plotting each point from scratch. They also help us quickly determine key features of the graph, such as where it begins, its vertex, and changes in its symmetry.

Horizontal Shift

A horizontal shift occurs when the graph of a function is moved left or right on the coordinate plane. In our example, the function \( f(x) = |x + 3.5| \) represents this type of transformation. To understand the shift direction, notice the expression inside the absolute value: \( x + 3.5 \). In fact, this can be rewritten as \( x - (-3.5) \), indicating a shift to the left by 3.5 units.

• The general rule is: if you have \( |x - c| \), the graph shifts right by \( c \) units if \( c > 0 \).
• If you have \( |x + c| \), treat it as \( |x - (-c)| \), meaning a left shift by \( c \) units.

Understanding this rule helps to visualize how transformations apply before ever graphing the equation. The entire V-shaped graph of \( h(x) = |x| \) moves horizontally, keeping its shape intact.

Vertical Shift

Vertical shifts occur when the graph moves up or down along the y-axis. This is done by adding or subtracting a constant outside the absolute value expression. For the function \( g(x) = |x| + 3.5 \), we observe a vertical shift upwards. The constant \( +3.5 \) indicates that the whole graph of \( h(x) = |x| \) is moved up by 3.5 units.

• For a positive \( +a \), the graph shifts up by \( a \) units.
• For a negative \( -a \), the graph shifts down by \( a \) units.

This type of transformation does not alter the symmetry or the V shape of the graph, but it does change the y-intercept and move all points higher (or lower) by the specified amount. Recognizing vertical shifts allows us to determine the new origin of the graph and any changes in its intercepts. In our example, the shift doesn't distort the graph of \( h(x) = |x| \), but raises its position upward uniformly.