https://file.aiquickdraw.com/mathbot/file/1732698068996iku52bvr.jpg

Let's break this down into two separate problems:

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### 1. **Describing the Transformation:**

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

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

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

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### 2. **Finding the New Function:**

You are asked to transform \( f(x) = x^7 \) 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 \( f(x) \), resulting in \( -x^7 \).
2. Shifting to the right by 1 unit replaces \( x \) with \( (x - 1) \), giving \( -(x - 1)^7 \).

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

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

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### 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 \( f(x) = x^2 \) into \( g(x) = -x^2 + 3 \)?
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?

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

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.

Understand how to transform functions, including reflections over the x-axis and vertical and horizontal shifts. Learn step-by-step how f(x) = 2x becomes g(x) = -2x - 5, and how f(x) = x^7 flips and shifts.

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