Math Snap

STEP 1

1. We are dealing with cubic functions, which have the general form $y = x^3$.
2. The transformations involve horizontal and vertical shifts.
3. The parameter $a$ in part (a) represents a horizontal shift.
4. The parameter $b$ in part (b) represents a vertical shift.

STEP 2

1. Analyze and sketch the graph for $y = (x+a)^3$.
2. Analyze and sketch the graph for $y = x^3 - b$.

STEP 3

The base function is $y = x^3$, which is a cubic function with an inflection point at the origin (0,0).

STEP 4

For $y = (x+a)^3$, the graph of $y = x^3$ is shifted horizontally to the left by $a$ units because $a > 0$.

STEP 5

Sketch the graph of $y = (x+a)^3$:
- Start with the graph of $y = x^3$.
- Shift every point on this graph $a$ units to the left.
- The inflection point moves from (0,0) to (-a,0).

STEP 6

The base function is again $y = x^3$.

STEP 7

For $y = x^3 - b$, the graph of $y = x^3$ is shifted vertically downward by $b$ units because $b > 0$.

Sketch the graph of $y = x^3 - b$:
- Start with the graph of $y = x^3$.
- Shift every point on this graph $b$ units downward.
- The inflection point moves from (0,0) to (0,-b).
The graphs for each function have been sketched with the described transformations.