Problem 41 - Precalculus with Limits

Question

In Exercises 19-42, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. \(f(x) = |x+4|\)

Step-by-Step Solution

Answer

A graph of the function \(f(x) = |x+4|\) should depict the typical 'V' shape of the absolute value function, but shifted 4 units to the left on the x-axis. An appropriate viewing window would range from \(x=-10\) to \(x=10\) and \(y=0\) to \(y=10\).

Step 1: Understanding the transformation

Recognize the function \(f(x) = |x+4|\) is a transformation of the fundamental absolute function \(|x|\). The '+4' inside the absolute value means the graph of the function will be shifted 4 units to the left.

Step 2: Graph the function

Plug the function \(f(x) = |x+4|\) into a graphing utility. Ensure that the vertical axis (y-axis) intersects the graph at \(y=0\) and the horizontal axis (x-axis) intersects the graph at \(x=-4\) to depict the shift.

Step 3: Choose an appropriate viewing window

An appropriate viewing window for this graph would include \(x\) values from -10 to 10 and \(y\) values from 0 to 10. This range can fully depict the transformation and typical 'V' shape of the absolute value function.

Key Concepts

Function Transformations

Function transformations modify the appearance of a basic graph by shifting, stretching, compressing, or reflecting it. Let's focus on understanding how it works using absolute value functions as the base example. When you see a function like \( f(x) = |x+4| \), this is a transformation of the simpler absolute value function \( g(x) = |x| \). Here’s a breakdown of common transformations:

• Horizontal Shifts: If you add or subtract a value within the absolute value brackets, it moves the graph along the x-axis. For \( f(x) = |x+4| \), adding 4 shifts the graph 4 units to the left.
• Vertical Shifts: When numbers are added or subtracted outside the absolute value brackets, the entire graph moves up or down vertically.
• Reflections and Stretches: If there are multiplying factors, they can flip the graph over an axis or elongate/contract it.

The key to function transformations is recognizing these shifts and adjustments. It helps you predict and graph new functions based on simple changes.

Absolute Value Property

The absolute value property focuses on the distance a number is from zero on a number line. The graph of an absolute value function, such as \( y = |x| \), typically displays a 'V' shape. The expression \( |x+4| \) means all values within will be treated as non-negative.When you take any number and apply the absolute value, it outputs the distance from zero, thus always yielding a non-negative result. Hence, no part of the graph of \( f(x) = |x+4| \) will dip below the x-axis since absolute values are never negative.For graphing purposes:

• The vertex, or the point where the direction of the graph changes, is a crucial feature. For \( f(x) = |x+4| \), this vertex is at \( x = -4 \) due to the shift.
• The slope on each side of the vertex is similar in absolute value graphs. Lines connecting through the vertex symmetrically project on either side.

This property ensures absolute value graphs always show a symmetry around their vertex.

Graphing Utilities

Graphing utilities are tools that assist in visualizing functions like \( f(x) = |x+4| \). These can include software, calculators, or other digital tools that aid in plotting the function to examine its behavior and transformations.When utilizing graphing utilities:

• Input the Function: Begin by carefully entering \( f(x) = |x+4| \) as directed by the tool you are using.
• Select Appropriate Window: Adjust the viewable section of the graph. For \( f(x) = |x+4| \), a window showing x-values from -10 to 10 and y-values from 0 to 10 allows you to observe the function comfortably.
• Interpreting Output: Look for the 'V' shape of the graph and confirm that it centers around coordinate \( (-4, 0) \), visualizing the transformation from \( g(x) = |x| \).

Using these utilities can streamline the process of understanding and sketching graphs, making mathematical concepts more approachable and easier to grasp.