Question

y equals x cubed minus 2.

how to graph

Answers

GPT-4o mini
To graph the function \( y = x^3 - 2 \), follow these steps:

### Step 1: Understand the Function
The function \( y = x^3 - 2 \) is a cubic function. It has a shape characteristic of cubic equations, and the \(-2\) indicates that the entire graph is shifted down by 2 units.

### Step 2: Determine Key Features
1. **Intercepts**:
- **Y-intercept**: Set \( x = 0 \):
\[
y = 0^3 - 2 = -2
\]
So, the y-intercept is at (0, -2).

- **X-intercepts**: Set \( y = 0 \):
\[
0 = x^3 - 2 \implies x^3 = 2 \implies x = \sqrt[3]{2} \approx 1.26
\]
So, the x-intercept is at \( \left(\sqrt[3]{2}, 0\right) \approx (1.26, 0) \).

2. **End Behavior**:
- As \( x \to -\infty \), \( y \to -\infty \).
- As \( x \to +\infty \), \( y \to +\infty \).

3. **Symmetry**: Cubic functions are not symmetric to the y-axis or x-axis. However, they may exhibit rotational symmetry about the origin.

### Step 3: Calculate Additional Points
Choose several values of \( x \) and calculate the corresponding \( y \) values to get points for the graph:

- \( x = -2 \):
\[
y = (-2)^3 - 2 = -8 - 2 = -10 \quad \Rightarrow \quad (-2, -10)
\]

- \( x = -1 \):
\[
y = (-1)^3 - 2 = -1 - 2 = -3 \quad \Rightarrow \quad (-1, -3)
\]

- \( x = 1 \):
\[
y = 1^3 - 2 = 1 - 2 = -1 \quad \Rightarrow \quad (1, -1)
\]

- \( x = 2 \):
\[
y = 2^3 - 2 = 8 - 2 = 6 \quad \Rightarrow \quad (2, 6)
\]

### Step 4: Plot the Points
Plot the points obtained on the graph:
- \( (-2, -10) \)
- \( (-1, -3) \)
- \( (0, -2) \) (y-intercept)
- \( (1, -1) \)
- \( \left(\sqrt[3]{2}, 0\right) \approx (1.26, 0) \) (x-intercept)
- \( (2, 6) \)

### Step 5: Draw the Curve
Connect the points smoothly. The graph will pass through the plotted points, starting from the lower left (downward as it approaches \( -\infty \)), rising through the x-intercept, and continuing upward to the right as it approaches \( +\infty \).

### Step 6: Label the Axes
Ensure to label the x-axis and y-axis, and if possible, mark the intercepts on the graph for clarity.

### Final Graph
The graph of \( y = x^3 - 2 \) will have the general shape of a cubic curve, with a point of inflection and the specified key features you’ve plotted.

If you would like a visual representation but cannot create it, graphing software or graphing calculators can help illustrate this function.

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