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Chapter 3: Problem 17

Describe how the graph of the function is a transformation of the graph of the original function \(f.\) $$y=f(x)-7$$

Short Answer

Expert verified
The graph of the function is shifted downward by 7 units.

Step by step solution

01

Understanding the Original Function

First, consider the original function, typically written as \( y = f(x) \). This serves as the foundational graph before any transformations are applied.
02

Identifying the Transformation Type

The transformation indicated in the function \( y = f(x) - 7 \) suggests a 'vertical shift'. The mathematical operation performed outside the function (i.e., subtraction) directly affects the graph's range.
03

Applying the Vertical Shift

Subtracting 7 from \( f(x) \) implies that every point on the graph of \( y = f(x) \) is shifted downward by 7 units along the y-axis. This means if a point on the original graph is at \( (a, b) \), it will move to \( (a, b-7) \) on the transformed graph.
04

Describing the Graph Transformation

The transformation does not stretch or compress the graph; it simply moves it down. Therefore, the shape remains unchanged, only its position along the vertical axis is altered.

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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 the graph of a function involves moving the graph up or down along the y-axis without altering its shape. When considering a function in the form of \( y = f(x) - 7 \), the '-7' indicates a downward vertical shift of 7 units.

This happens because the subtraction occurs outside the function itself, affecting all y-values of the graph uniformly.
  • Every point on the original graph, represented as \( (a, b) \), relocates to \( (a, b-7) \) after the shift.
  • Visualizing this, you see the whole graph descending 7 units from its initial position.
  • The graph's vertical shift does not modify the graph's overall shape or orientation; only its vertical position changes.
Understanding vertical shifts helps in quickly identifying how transformations affect a function visually.
Graph of a Function
The graph of a function provides a visual representation of the relationship between variables, typically expressed as \( y = f(x) \). This serves as the starting point before any transformations are applied.

To grasp how transformations like vertical shifts affect it, consider:
  • The x-axis represents all possible inputs or domain values, while the y-axis reflects the corresponding output or range values.
  • Points plotted on the graph demonstrate the function's behavior and demonstrate how y-values change with different x-values.
  • The original graph serves as the foundation for evaluating transformations, allowing comparisons before and after changes are made.
Understanding the graph of a function is fundamental as it offers a frame of reference to distinguish transformations.
Algebraic Transformations
Algebraic transformations involve changes to the equation of a function which subsequently alter its graph. These transformations modify the position, shape, size, or orientation of the graph.

For this exercise, our primary focus is on a particular type of transformation:
  • **Vertical Shift**: This transformation is apparent when either a constant is added to or subtracted from the function. For example, in \( y = f(x) - 7 \), a downward shift occurs due to subtraction outside the function.
  • Algebraic transformations like vertical shifts are crucial in quickly modifying graphs for solving real-world problems, especially when understanding and analyzing functional relationships visually and mathematically.
  • Algebraic transformations are not limited to vertical shifts; they can also include horizontal shifts, reflections, stretches, and compressions depending on the type of operation applied within or outside the function.
By mastering these transformations, you can better interpret and manipulate graphs across various mathematical contexts.

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Most popular questions from this chapter

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