Question

graph y equals x cubed minus 4.

Answers

GPT-4o mini
To graph the function \( y = x^3 - 4 \), follow these steps:

1. **Identify Key Features**:
- **Shape**: This is a cubic function, which generally has an "S" shape. As \( x \) approaches positive or negative infinity, \( y \) also approaches infinity in both directions.
- **Y-intercept**: Set \( x = 0 \):
\[
y = 0^3 - 4 = -4
\]
This means the graph crosses the y-axis at (0, -4).
- **X-intercepts**: Set \( y = 0 \):
\[
0 = x^3 - 4
\]
\[
x^3 = 4 \implies x = \sqrt[3]{4} \approx 1.587
\]
This means there is one real root at approximately \( x \approx 1.587 \).

2. **Plot Points**:
To better understand the shape of the graph, calculate some additional points:
- \( x = -2 \):
\[
y = (-2)^3 - 4 = -8 - 4 = -12 \quad \text{(point: (-2, -12))}
\]
- \( x = -1 \):
\[
y = (-1)^3 - 4 = -1 - 4 = -5 \quad \text{(point: (-1, -5))}
\]
- \( x = 1 \):
\[
y = 1^3 - 4 = 1 - 4 = -3 \quad \text{(point: (1, -3))}
\]
- \( x = 2 \):
\[
y = 2^3 - 4 = 8 - 4 = 4 \quad \text{(point: (2, 4))}
\]

3. **Sketch the Graph**:
Using the points calculated, you can draw the graph:
- Start from the left, at \( (-2, -12) \) going up to \( (-1, -5) \).
- Continue through the point \( (0, -4) \) and then to \( (1, -3) \).
- The function will continue to rise and pass through the x-intercept at \( (\sqrt[3]{4}, 0) \) before moving up to \( (2, 4) \) and beyond.

### Graphical Representation:

While I can't physically draw, you can create a sketch based on the points listed above to visualize the shape of the graph as it goes from the bottom left to the top right, showing the "S" curve typical of cubic functions.

### Summary:
- Y-intercept: (0, -4)
- X-intercept: Approximately (1.587, 0)
- Several points help shape the graph: (-2, -12), (-1, -5), (1, -3), (2, 4).

The overall behavior is increasing as \( x \) moves away from the origin in both directions.

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