compare the graph of each function with graph f(x)=x^3     a.g(x)=x^3-1   b.h(x)=(x-1)^3    c.k(x)=(x+2)^3+1

Let's analyze how the graphs of the given functions compare to the graph of \( f(x) = x^3 \). The graph of \( f(x) = x^3 \) is a basic cubic function, symmetric around the origin, with a characteristic S-shape. The key transformations are vertical shifts, horizontal shifts, and translations.

### a. \( g(x) = x^3 - 1 \)
- **Transformation:** This function subtracts 1 from \( f(x) = x^3 \). 
- **Effect:** This represents a **vertical shift downward** by 1 unit. Every point on the graph of \( f(x) = x^3 \) is shifted 1 unit lower. The shape of the graph remains the same, but the entire graph moves down.
- **Graph comparison:** The turning point (inflection point) of \( f(x) \) at (0, 0) is moved to (0, -1).

### b. \( h(x) = (x - 1)^3 \)
- **Transformation:** This involves replacing \( x \) with \( x - 1 \). 
- **Effect:** This is a **horizontal shift to the right** by 1 unit. The graph of \( f(x) = x^3 \) is moved 1 unit to the right. Again, the shape of the graph is preserved.
- **Graph comparison:** The inflection point at (0, 0) for \( f(x) \) is now at (1, 0).

### c. \( k(x) = (x + 2)^3 + 1 \)
- **Transformation:** This involves both replacing \( x \) with \( x + 2 \) and adding 1 to the function.
- **Effect:** This is a **horizontal shift to the left** by 2 units (due to \( x + 2 \)) and a **vertical shift upward** by 1 unit (due to \( +1 \)). The graph is translated to the left and up, but the shape is unchanged.
- **Graph comparison:** The inflection point at (0, 0) is now moved to (-2, 1).

### Summary of the Transformations:
- \( g(x) = x^3 - 1 \): **Shifted down** 1 unit.
- \( h(x) = (x - 1)^3 \): **Shifted right** 1 unit.
- \( k(x) = (x + 2)^3 + 1 \): **Shifted left** 2 units and **shifted up** 1 unit.

Would you like a detailed graph for better visualization or further explanations on any transformations?

### 5 Questions:
1. What is the inflection point of the graph \( g(x) = x^3 - 1 \)?
2. How does the graph of \( f(x) = x^3 \) compare to \( h(x) = (x - 1)^3 \) at large values of \( x \)?
3. Why does \( k(x) = (x + 2)^3 + 1 \) shift both horizontally and vertically?
4. How does the slope of \( f(x) = x^3 \) change under the transformations?
5. What would be the transformation if you have \( m(x) = (x - 3)^3 + 2 \)?

### Tip:
Horizontal shifts affect the input variable \( x \), while vertical shifts affect the output or function value \( f(x) \).

Compare the graph of each function with graph f(x)=x^3. a. g(x)=x^3-1 b. h(x)=(x-1)^3 c. k(x)=(x+2)^3+1

Explore the transformations of the cubic function f(x)=x^3 by comparing it to g(x)=x^3-1, h(x)=(x-1)^3, and k(x)=(x+2)^3+1. Learn about vertical and horizontal shifts, inflection points, and how these changes affect the graph of cubic functions. Suitable for students in Grades 9-12.

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