Problem 43 - Precalculus with Limits

Question

In Exercises 25-54, \(g\) is related to one of the parent functions described in Section 1.6. (a) Identify the parent function \(f\). (b) Describe the sequence of transformations from \(f\) to \(g\). (c) Sketch the graph of \(g\). (d) Use function notation to write \(g\) in terms of \(f\). \(g (x) = - |x + 4| + 8\)

Step-by-Step Solution

Answer

The parent function is the absolute value function \(f(x) = |x|\). The transformations from \(f(x)\) to \(g(x)\) are the reflection over the x-axis, a horizontal shift four units to the left, and a vertical shift eight units upward. The graph of \(g(x)\) is an inverted 'V' shape shifted four units left and eight units up. So \(g(x)\) is \(g(x) = -f(x + 4) + 8\)

Step 1: Identifying the Parent Function

The parent function \(f(x)\) is the absolute value function \(f(x) = |x|\). This is identified by the presence of absolute value bars in the definition of \(g(x)\). An absolute value function returns the non-negative value of its input, determining the basic 'V' shape in its graph.

Step 2: Describing the Transformations

The transformations from \(f(x)\) to \(g(x)\) have been given by \(g(x) = -|x + 4| + 8\). So, there are three transformations applied to \(f(x)\): \n1. A reflection about x-axis when the function is multiplied by -1.\n2. A horizontal shift four units to the left due to the addition of 4 inside the absolute value.\n3. A vertical shift eight units upwards due to the addition of 8 outside the absolute value.

Step 3: Sketching the Graph

The graph of \(g(x)\) starts as the 'V' shape of the absolute value function \(f(x) = |x|\), but is reflected to open downwards due to being multiplied by -1. This inverted 'V' is then shifted four units to left on the x-axis and eight units upwards on the y-axis.

Step 4: Writing g in Terms of f

After identifying the transformations, \(g(x)\) can now be written in terms of \(f(x)\) as follows: \(g(x) = -f(x + 4) + 8\).

Key Concepts

Understanding Parent Functions

Parent functions lay the groundwork for understanding a variety of other functions. These basic functions represent the simplest form of a given type.
They serve as the "parent" from which more complex versions can be developed by applying different transformations.
For example, the absolute value function, typically written as \( f(x) = |x| \), is one of these parent functions. It forms the foundational 'V' shape which is centered at the origin (0,0) and rises up symmetrically on both sides.

• The graph is characterized by a sharp corner at the origin.
• It reflects the idea that the output will always be non-negative, regardless of the sign of the input.

This understanding helps us manipulate and transform other functions by using the properties of the parent functions.

Exploring the Absolute Value Function

The absolute value function is a staple in algebra that simplifies handling both positive and negative inputs. By definition, it returns the non-negative value of whichever input it receives.
When we see the absolute value bars \(|x|\), it signals a reflection of negative input values over the y-axis to become positive. This is why the resulting graph forms a 'V' shape.

• The vertex is located at the origin, forming a symmetrical pattern.
• Each arm of the 'V' extends infinitely with a slope of 1 on either side of the y-axis.

In transformations, this function is crucial. For instance, applying operations inside and outside the bars can shift, stretch, or reflect this fundamental shape.

Graphing Transformations and Their Impact

Graphing transformations are techniques used to alter the position and orientation of a graph on a coordinate plane.
By applying transformations, we can move graphs around, flip them, or stretch and compress them.
Using \(g(x) = -|x + 4| + 8\) as an example:

• Reflection: The minus sign before \(|x|\) inverts the graph, flipping the 'V' shape upside down, indicating reflection across the x-axis.
• Horizontal Shift: The expression \(x + 4\) inside the bars shifts the graph left by 4 units. The positive 4 causes this shift contrary to intuition.
• Vertical Shift: Adding 8 after the absolute value moves the entire graph upwards by 8 units, lifting the inverted 'V'.

Such transformations allow us to precisely tailor graphs to specific equations, making visualization and understanding much easier.