Math Problem Statement

Describe the transformations for the function f(x) = -|x - 2| + 3 based on the graph provided.

Solution

The problem asks to describe the transformations for the function f(x)=x2+3f(x) = -|x - 2| + 3. Let's break this down:

Function: f(x)=x2+3f(x) = -|x - 2| + 3

  1. Start with the basic absolute value function: y=xy = |x|

    • This is a V-shaped graph with its vertex at the origin (0,0)(0,0).
  2. Horizontal shift (inside the absolute value): x2|x - 2|

    • The function y=x2y = |x - 2| shifts the graph to the right by 2 units.
    • The new vertex of the V-shape is now at (2,0)(2, 0).
  3. Vertical reflection (negative sign in front): x2-|x - 2|

    • The negative sign reflects the graph over the x-axis. So, instead of opening upwards, the V-shape now opens downwards, creating an inverted V.
  4. Vertical shift (addition outside): x2+3-|x - 2| + 3

    • The graph is then shifted up by 3 units, moving the vertex from (2,0)(2, 0) to (2,3)(2, 3).

Summary of Transformations:

  1. Shift right by 2 units.
  2. Reflect over the x-axis.
  3. Shift up by 3 units.

Looking at the graph in your image, the inverted V-shape with its vertex at (2,3)(2,3) is consistent with these transformations.


Would you like more details on any part of this explanation? Here are five related questions to further explore this concept:

  1. How would the graph change if the transformation was f(x)=x+23f(x) = |x + 2| - 3?
  2. What would the effect of a vertical stretch or compression (e.g., f(x)=2x2+3f(x) = 2|x - 2| + 3) be?
  3. How can you algebraically find the vertex of an absolute value function like f(x)=xh+kf(x) = |x - h| + k?
  4. How do reflections and shifts affect the symmetry of a function?
  5. How would you determine the domain and range of f(x)=x2+3f(x) = -|x - 2| + 3?

Tip: When analyzing transformations, always handle horizontal shifts (inside the function) before vertical shifts (outside).

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Math Problem Analysis

Mathematical Concepts

Absolute Value Functions
Graph Transformations

Formulas

f(x) = a|x - h| + k

Theorems

Transformation Theorems: Shifts, Reflections

Suitable Grade Level

Grades 9-12