Problem 31 - 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 \(f\). \(g (x) = 2 - (x + 5)^2\)

Step-by-Step Solution

Answer

The parent function is \(f(x)=x^2\). The graph of \(g(x) = 2 - (x + 5)^2\) is a parabola that has been reflected across the x-axis, shifted 5 units to the left, and shifted 2 units up. It can be described in terms of \(f(x)\) as \(g(x) = 2 - f(x + 5)\).

Step 1: Identifying the Parent Function

For the function \(g(x) = 2 - (x + 5)^2\), the parent function is \(f(x) = x^2\). This can be deduced by looking at the basic structure of the given function

Step 2: Describe the transformations

The function \(g (x) = 2 - (x + 5)^2\), has undergone a couple of transformations. From the original function, the graph has been reflected on the x-axis, this is evident from the negative sign present in front of the \(x^2\). Secondly, the graph has shifted or translated 5 units to the left, you get this from \(-5\) present in the bracket which implies a shift to the left by 5 units due to the negative sign. Lastly, the graph has also been shifted upwards by 2 units as shown by the \(+2\) outside the bracket. This tells us the graph has been shifted up by two units.

Step 3: Writing g In Terms of f

The function \( g(x) \) can be represented in terms of \( f(x) \) as \( g(x) = 2 - f(x + 5) \)

Key Concepts

Parent Function

Understanding the concept of a "parent function" is essential when learning about function transformations. A parent function is the simplest form of a set of functions that form a family. In this case, the given function is quadratic, and its parent function is the basic quadratic function:

• \(f(x) = x^2\)

The graph of this function is a parabola that opens upwards, centered at the origin (0,0). Recognizing the parent function helps identify what type of graph you will be transforming. Knowing the parent function is your starting roadmap in understanding how the function will change through various transformations.

Quadratic Function

Quadratic functions, such as

• \( f(x) = x^2 \)

are a crucial concept in algebra. These functions create graphs that are parabolic shapes, which may open upwards or downwards. The general form of a quadratic function is:

• \( f(x) = ax^2 + bx + c \)

In this case, our parent function (

• \(f(x) = x^2\)

) is the most basic form where \(a = 1\), \(b = 0\), and \(c = 0\). The function represents a symmetrical curve about the y-axis, allowing for easy analysis when performing transformations.

Graphing Transformations

Graphing transformations involve altering the position or shape of a graph in a systematic way. For

• \(g(x) = 2 - (x + 5)^2\)

there are several transformations from the parent function \(f(x) = x^2\):

• Reflection: The graph is reflected over the x-axis because of the negative sign in
  • \(-(x + 5)^2\)
• Horizontal Translation: The "+5" inside the squared term shifts the graph 5 units to the left. This might seem counterintuitive with the positive sign, but inside the parenthesis, it shows a leftward shift.
• Vertical Transformation: The "+2" outside the transformation translates the whole graph 2 units up.

These transformations alter the shape and location of the graph, giving a transformed version of the initial parabola defined by the parent function.

Reflections and Translations

Reflections and translations are subsets of graphing transformations that adjust how the graph appears on the coordinate plane. When a function is reflected, it is mirrored across a certain axis. In the function

• \(g(x) = 2 - (x + 5)^2\)

the reflection over the x-axis flips the parabola upside down.Translations involve shifting a graph vertically or horizontally. In our example:

• Horizontal Translation: Occurs inside parentheses, meaning \( (x + 5) \) results in a 5 unit leftward shift.
• Vertical Translation: The "+2" causes the parabola to rise 2 units.

Through these movements, you can manipulate the function to see how graphs transform geometrically. By practicing these concepts, graphing becomes intuitive and dynamic, reinforcing spatial comprehension.