Math Problem Statement

A point A(1, -4) lies on the graph of the function y = 3 - f(x). Determine the coordinates of point A on the graph of the transformed function y = f(x - 2).

Solution

The question provided is written in Persian and includes a mathematical expression. I'll translate and interpret it to guide you through the solution.

Translation:

  • The problem gives the point A(1,4)A(1, -4) on the graph of the function y=3f(x)y = 3 - f(x). It asks you to determine the coordinates of point AA on the graph of the function y=f(x2)y = f(x - 2).

Step-by-Step Solution:

  1. Initial Function: The given function is y=3f(x)y = 3 - f(x), and point A(1,4)A(1, -4) lies on this curve.

    • This means that for x=1x = 1, y=4y = -4.
    • So, we substitute x=1x = 1 and y=4y = -4 into the equation y=3f(x)y = 3 - f(x): 4=3f(1)-4 = 3 - f(1) Solving for f(1)f(1): f(1)=3+4=7f(1) = 3 + 4 = 7 Therefore, f(1)=7f(1) = 7.
  2. New Function: The next part of the problem asks for the coordinates on the graph of the transformed function y=f(x2)y = f(x - 2).

    • In this case, we have a horizontal shift of the function by 2 units to the right.
    • We know f(1)=7f(1) = 7, so now we are interested in the value of f(x2)f(x-2) when x=1x = 1.
  3. Finding f(x2)f(x - 2): If f(1)=7f(1) = 7, and we are now dealing with f(x2)f(x - 2), we substitute x=3x = 3 (since x2=1x - 2 = 1) into f(x2)f(x - 2):

    • For x=3x = 3, f(32)=f(1)=7f(3 - 2) = f(1) = 7.

Thus, the coordinates on the new graph of y=f(x2)y = f(x - 2) corresponding to point A(1,4)A(1, -4) are A(3,7)A(3, 7).

Would you like more details or have any further questions?


Here are 5 related questions to extend this concept:

  1. What is the effect of a vertical shift on the graph of f(x)f(x)?
  2. How do horizontal shifts affect the domain of a function?
  3. What happens to the graph of f(x)f(x) if you apply both vertical and horizontal shifts?
  4. Can we find the inverse of f(x)f(x), and how does the graph change?
  5. What are the coordinates of point AA if the function is transformed to f(2x)f(2x)?

Tip: Always check how transformations like horizontal shifts impact the input variable xx when applying them to functions.

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Math Problem Analysis

Mathematical Concepts

Function transformations
Graph shifts

Formulas

f(x - h) results in a horizontal shift by h units
y = 3 - f(x)

Theorems

Graph transformations
Horizontal and vertical shifts

Suitable Grade Level

Grades 10-12