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Chapter 1: Problem 6

Describe how each function is a transformation of the original function \(f(x)\) $$ f(x)+8 $$

Short Answer

Expert verified
This is a vertical shift upward by 8 units.

Step by step solution

01

Identify the Transformation

The expression is given as new function: \( f(x) + 8 \). This represents a transformation of the original function \( f(x) \).
02

Determine the Type of Transformation

Adding a constant value directly to a function \( f(x) \) represents a vertical shift. Here, the original function is shifted vertically.
03

Perform the Transformation

For the function \( f(x) + 8 \), each point on the graph of \( f(x) \) is moved up by 8 units. This is a vertical upward shift of the graph of the function \( f(x) \).
04

Interpret the Results

By shifting every point of \( f(x) \) up 8 units, the new function is obtained. The shape of the graph remains the same, only its position on the y-axis changes.

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Key Concepts

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

Vertical Shift
A vertical shift in function transformations occurs when a constant number is added or subtracted from the function. In our example, the function is expressed as \( f(x) + 8 \). This indicates that the entire graph of the function \( f(x) \) is shifted upwards by 8 units.
  • An upward shift happens when a positive number is added. For \( f(x) + 8 \), it moves the graph 8 units up.
  • A downward shift occurs when a negative number is subtracted. For instance, in \( f(x) - c\), the graph moves down by \( c \) units if \( c\) is positive.
The key point to remember is that vertical shifts do not distort the shape of the graph. The entire function maintains its original form; only its location on the y-axis changes.
Graphical Interpretation
When interpreting transformations graphically, it's crucial to understand how the shift affects the graph's appearance. In the case of a vertical shift like \( f(x) + 8 \):
  • Every point on the graph moves vertically upwards by 8 units.
  • This change affects the y-coordinates of each point, while x-coordinates remain unchanged.
Visually, imagine taking the entire graph and sliding it straight up without any additional scaling or skewing. It's important to note that, though the position of the graph changes, its core features鈥攕uch as intercepts with the x-axis (if applicable) and the overall trajectory鈥攔emain unaltered. Understanding this will help in recognizing the effects of similar transformations applied to other functions.
Function Notation
Function notation is a concise way to represent mathematical functions. It not only indicates which operations are being performed but also guides us through transformations.
  • The function \( f(x) \) represents a particular mathematical rule or equation.
  • Adding a constant, like in \( f(x) + 8 \), results in a simple and clear adjustment to the original function.
This notation allows you to see quickly that just a vertical shift is occurring鈥攚ithout affecting how the input x-values are processed. In general, function notation is very powerful for expressing transformations:
  • You can easily differentiate between transformations like vertical shifts \( (f(x) + c) \), horizontal shifts \( (f(x-c)) \), and others.
Using function notation efficiently provides a foundation to understand complex transformations with ease.

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Most popular questions from this chapter

Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ g(x)=5(x+3)^{2}-2 $$

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