Problem 9 - Algebra 2

Question

For each function, identify the translation of the parent function. Then graph the function. \(y=|x+5|\)

Step-by-Step Solution

Answer

The parent function is \(y = |x|\). The given function \(y = |x+5|\) has been translated 5 units to the left. The vertex is now at (-5,0).

Step 1: Identify the Parent Function

The parent function of the given function is the absolute value function, expressed as \(y = |x|\). This is a basic function that forms a 'V' shape, with its vertex at the origin (0,0) and opens upwards.

Step 2: Determine the Translation

To identify how the parent function has been translated, compare the given function \(y = |x + 5|\) to the parent function \(y = |x|\). The \(+5\) inside the absolute value indicates a horizontal shift to the left by 5 units since it is added to the input \(x\). There is no vertical translation as there is no constant added outside of the absolute value.

Step 3: Graph the Translated Function

To graph the function \(y = |x + 5|\), start with the graph of the parent function \(y = |x|\). Then, translate the graph 5 units to the left. Plot key points from the parent function, shifted 5 units left, such as (-5,0), to reflect the translation. Ensure that after translation, the graph maintains the same 'V' shape and opens upwards, just shifted horizontally.

Key Concepts

Parent Function

Understanding the parent function is essential when starting to work with absolute value equations. The parent function for an absolute value equation is typically written as
\( y = |x| \)
This function produces a V-shaped graph symmetric about the y-axis, with its vertex at the origin of the coordinate plane. The graph opens upwards and its slopes are equal and opposite on either side of the vertex. The importance of recognizing the parent function lies in its role as the starting point for any transformations such as shifts, stretches, or reflections that may occur in more complex absolute value functions.

Translations of Functions

Translations of functions involve sliding the entire graph of a function either vertically, horizontally, or both without changing its shape or orientation. In the given exercise, \( y = |x + 5| \),
it's evident that there is a horizontal translation as indicated by the
\( +5 \)
inside the absolute value brackets. This tells us that each point on the parent function's graph will move 5 units to the left. It's crucial to note that if a negative value were inside the brackets, we would translate the graph to the right instead. Understanding translations is a foundational skill in graphing as it allows us to predict and accurately graph the behavior of altered functions.

Graphing Techniques

Graphing a transformed function involves several key techniques. Firstly, it is important to know the shape and position of the parent function's graph. From there, you can use a table of values to find key points that you'll then shift according to the transformation.
In our case, the graph of \( y = |x| \)
is shifted 5 units left to graph \( y = |x + 5| \).
You would plot key points from the parent function, now moved 5 units to the left, and then draw the V-shaped graph resembling the parent function's shape. These techniques are a combination of understanding translations and the effect on the graph's orientation.

Absolute Value Transformations

When dealing with absolute value transformations, recognizing the changes inside and outside of the absolute value brackets is vital. Changes within the brackets, like the \( +5 \)
in \( y = |x + 5| \),
affect the horizontal position of the graph. Any value added or subtracted outside the absolute value symbols affects the vertical position. Apart from translations, absolute value functions can also undergo stretches or compressions, and reflections. The key to mastering absolute value transformations is to apply each change in a step-by-step manner to ensure accuracy in the resulting graph.