Math Problem Statement

The graph of the function f(x) is provided, and the task is to find values of b and d for the transformation g(x) = f(x - b) + d.

Solution

The problem presents a graph of the function f(x)f(x), and asks in part (a) to find the values of bb and dd for the transformation:

g(x)=f(xb)+dg(x) = f(x - b) + d

To determine these values:

  • bb: The parameter bb represents a horizontal shift. If f(x)f(x) is shifted to the right by bb, it moves the graph of f(x)f(x) to the right.
  • dd: The parameter dd represents a vertical shift. Adding dd to the function f(x)f(x) will shift the graph vertically up or down.

Analyzing the Graph:

The graph of f(x)f(x) starts at f(0)=2f(0) = 2, and the function appears piecewise linear, having distinct flat and sloped segments. We need to determine the corresponding transformation g(x)g(x) based on the shift.

  • If the graph of g(x)g(x) is obtained by shifting the graph of f(x)f(x) horizontally by bb and vertically by dd, then:
    • bb is the horizontal shift that translates the graph.
    • dd is the vertical shift.

Given the transformation formula g(x)=f(xb)+dg(x) = f(x - b) + d, to find bb and dd, you must compare how the graph is shifted from f(x)f(x) to g(x)g(x).


Would you like further clarification, or should I provide a more detailed breakdown of how to compute bb and dd? Here are 5 related questions to help understand transformations:

  1. What happens to the graph of a function if we shift it by f(xb)f(x - b)?
  2. How does the graph behave when we apply a vertical shift f(x)+df(x) + d?
  3. Can you identify specific points on f(x)f(x) to understand how the graph is transformed?
  4. How do horizontal and vertical shifts affect key points like intercepts and critical points?
  5. What real-world situations might involve similar function transformations?

Tip: Always start by identifying key points on the original graph to see how they shift after applying transformations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Graph transformations
Horizontal shifts
Vertical shifts

Formulas

g(x) = f(x - b) + d

Theorems

-

Suitable Grade Level

Grades 10-12