Question

Find seven ordered pairs to the equation y equals x squared minus 2. Then determine its graph.

Answer 1

To find ordered pairs for the equation \( y = x^2 - 2 \), we can choose several values for \( x \), calculate the corresponding \( y \) values, and then list the ordered pairs.

### Calculate Ordered Pairs
1. **For \( x = -3 \)**:
\[
y = (-3)^2 - 2 = 9 - 2 = 7 \quad \Rightarrow \quad (-3, 7)
\]

2. **For \( x = -2 \)**:
\[
y = (-2)^2 - 2 = 4 - 2 = 2 \quad \Rightarrow \quad (-2, 2)
\]

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

4. **For \( x = 0 \)**:
\[
y = 0^2 - 2 = 0 - 2 = -2 \quad \Rightarrow \quad (0, -2)
\]

5. **For \( x = 1 \)**:
\[
y = (1)^2 - 2 = 1 - 2 = -1 \quad \Rightarrow \quad (1, -1)
\]

6. **For \( x = 2 \)**:
\[
y = (2)^2 - 2 = 4 - 2 = 2 \quad \Rightarrow \quad (2, 2)
\]

7. **For \( x = 3 \)**:
\[
y = (3)^2 - 2 = 9 - 2 = 7 \quad \Rightarrow \quad (3, 7)
\]

### Ordered Pairs
The seven ordered pairs based on the calculations above are:
1. \( (-3, 7) \)
2. \( (-2, 2) \)
3. \( (-1, -1) \)
4. \( (0, -2) \)
5. \( (1, -1) \)
6. \( (2, 2) \)
7. \( (3, 7) \)

### Graph of the Function
To graph \( y = x^2 - 2 \), we can plot the points we calculated:

- \( (-3, 7) \)
- \( (-2, 2) \)
- \( (-1, -1) \)
- \( (0, -2) \)
- \( (1, -1) \)
- \( (2, 2) \)
- \( (3, 7) \)

### Characteristics of the Graph
- The graph is a parabola that opens upwards.
- The vertex of the parabola is at the point \( (0, -2) \).
- As \( x \) moves away from the vertex (in either direction), \( y \) increases.
- The graph is symmetric with respect to the y-axis since it is an even function.

In summary, the graph of \( y = x^2 - 2 \) is a parabola that passes through the points defined above, with its vertex located at \( (0, -2) \).