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Chapter 1: Problem 2

Describe how each function is a transformation of the original function \(f(x)\) $$ f(x+43) $$

Short Answer

Expert verified
\( f(x+43) \) shifts \( f(x) \) 43 units to the left.

Step by step solution

01

Identify the Original Function

The original function provided is \( f(x) \). This function is commonly referred to as the 'parent' or 'base' function.
02

Analyze the Transformation

The expression \( f(x+43) \) represents a horizontal transformation of the function \( f(x) \).
03

Determine the Type of Horizontal Transformation

The function \( f(x+43) \) involves adding 43 to the input \( x \) before applying the function \( f \). Such a transformation indicates a horizontal shift.
04

Specify the Direction and Magnitude of the Shift

Adding a positive number inside the function argument (i.e., within \( f(x+43) \)) results in a horizontal shift to the left. Therefore, \( f(x+43) \) represents a shift of 43 units to the left from \( f(x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Horizontal Shift
When it comes to function transformations, a horizontal shift is one of the key modifications you can make to a function. This transformation involves moving the graph of a function along the horizontal axis without changing its shape. In our example, the expression given is \( f(x+43) \). This means we are adding 43 to the input \( x \) before the function \( f(x) \) is applied.
  • Adding a positive number to \( x \) results in the graph shifting to the left.
  • If you were to subtract a number from \( x \), the graph would shift to the right.
In general, shifting left or right helps to redefine the starting point of the function on the graph. So for \( f(x+43) \), the entire graph of the function \( f(x) \) moves 43 units to the left.
Exploring the Parent Function
The parent function is the simplest form of a family of functions that maintains the core characteristics of that function family. In the problem given, \( f(x) \) is identified as the parent function as it represents the original, unaltered form.
  • A parent function serves as a baseline from which various transformations can be observed.
  • All function transformations, including vertical and horizontal shifts, begin from the parent function.
Maintaining understanding of the parent function is crucial, as it helps students identify the effects that transformations, such as horizontal shifts, have on the graph's structure.
Delving into Horizontal Transformation
Horizontal transformation involves any change to the function that affects the horizontal positioning of its graph. This can include shifts, stretches, or compressions along the horizontal axis. In our specific instance, \( f(x+43) \), the transformation is solely a horizontal shift.
  • Horizontal shifts do not change the shape of the graph; only its position along the x-axis.
  • Other forms of horizontal transformation can include horizontal stretches, which spread the graph wider or narrower.
Such transformations may seem subtle but have meaningful impacts on how a function behaves. Recognizing and predicting these changes is a valuable skill in understanding function dynamics.

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Most popular questions from this chapter

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