Math Problem Statement

Describe the transformation of the function f(x) = 2x to the function g(x) = -2x - 5. Find the function which flips f(x) = x^7 in the x-axis and then shifts it to the right by 1 unit.

Solution

Let's break this down into two separate problems:


1. Describing the Transformation:

You are asked to describe how the function f(x)=2xf(x) = 2x transforms to g(x)=2x5g(x) = -2x - 5.

Step-by-step transformation:

  1. Reflection over the x-axis: The coefficient of xx changes from 22 to 2-2, indicating a reflection over the x-axis.
  2. Vertical translation (shift): The entire function is shifted downward by 5 units due to the 5-5 term.

Summary: f(x)g(x)f(x) \to g(x) involves a reflection over the x-axis and a downward shift of 5 units.


2. Finding the New Function:

You are asked to transform f(x)=x7f(x) = x^7 by:

  1. Flipping it over the x-axis, and then
  2. Shifting it to the right by 1 unit.

Step-by-step transformation:

  1. Flipping over the x-axis changes the sign of f(x)f(x), resulting in x7-x^7.
  2. Shifting to the right by 1 unit replaces xx with (x1)(x - 1), giving (x1)7-(x - 1)^7.

Final function: The transformed function is g(x)=(x1)7g(x) = -(x - 1)^7.


Let me know if you'd like further clarification on these transformations!


Related Questions:

  1. What happens to a function when it is reflected over the y-axis?
  2. How do vertical and horizontal shifts affect the graph of a function?
  3. What transformations would change f(x)=x2f(x) = x^2 into g(x)=x2+3g(x) = -x^2 + 3?
  4. How can transformations be used to model real-world scenarios?
  5. How do you determine if a function transformation involves a stretch or compression?

Tip:

Always remember that transformations involving horizontal shifts (xh)(x - h) are opposite of what they appear (right if h>0h > 0, left if h<0h < 0). Vertical shifts +k+k and k-k behave intuitively!

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Math Problem Analysis

Mathematical Concepts

Transformations of functions
Reflections
Vertical and horizontal shifts

Formulas

g(x) = a * f(x) + k (for vertical transformations)
f(x - h) for horizontal transformations

Theorems

Function transformations properties

Suitable Grade Level

Grades 9-11