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 $f(x)$.