Math Problem Statement

A quadratic function y, equals, f, of, xy=f(x) is plotted on a graph and the vertex of the resulting parabola is left parenthesis, minus, 4, comma, 5, right parenthesis(−4,5). What is the vertex of the function defined as g, of, x, equals, f, of, minus, x, minus, 3g(x)=f(−x)−3?

Solution

To find the vertex of the function g(x)=f(x)3g(x) = f(-x) - 3, we start with the vertex of f(x)f(x), which is at (4,5)(-4, 5).

  1. The transformation f(x)f(-x) reflects the graph across the y-axis. Thus, the vertex of f(x)f(-x) becomes (4,5)(4, 5).
  2. The transformation 3-3 shifts the graph downward by 3 units. This moves the vertex from (4,5)(4, 5) to (4,53)=(4,2)(4, 5 - 3) = (4, 2).

Therefore, the vertex of g(x)g(x) is (4,2)(4, 2).

Would you like more details or do you have any questions? Here are some related questions for further exploration:

  1. How does the reflection across the y-axis affect the function's graph?
  2. What are the general effects of shifting a function vertically or horizontally?
  3. How can you derive the standard form of a quadratic function from its vertex?
  4. What are some real-life applications of quadratic functions?
  5. Can you explain the significance of the vertex in the context of a quadratic function?

Tip: Always note the transformations applied to functions, as they significantly impact the graph's features!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Transformations

Formulas

Vertex form of a quadratic function
Transformations of functions

Theorems

Vertex transformation for quadratic functions

Suitable Grade Level

Grades 8-10