Question

Find seven ordered pairs to the equation y equals x squared minus 3. Then determine its graph.

Answer 1

To find seven ordered pairs for the equation \( y = x^2 - 3 \), we can select different values of \( x \) and calculate the corresponding \( y \) values.

Let's calculate:

1. **For \( x = -3 \)**:
\[
y = (-3)^2 - 3 = 9 - 3 = 6 \quad \Rightarrow \quad (-3, 6)
\]

2. **For \( x = -2 \)**:
\[
y = (-2)^2 - 3 = 4 - 3 = 1 \quad \Rightarrow \quad (-2, 1)
\]

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

4. **For \( x = 0 \)**:
\[
y = (0)^2 - 3 = 0 - 3 = -3 \quad \Rightarrow \quad (0, -3)
\]

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

6. **For \( x = 2 \)**:
\[
y = (2)^2 - 3 = 4 - 3 = 1 \quad \Rightarrow \quad (2, 1)
\]

7. **For \( x = 3 \)**:
\[
y = (3)^2 - 3 = 9 - 3 = 6 \quad \Rightarrow \quad (3, 6)
\]

Now we have the following seven ordered pairs:

1. \( (-3, 6) \)
2. \( (-2, 1) \)
3. \( (-1, -2) \)
4. \( (0, -3) \)
5. \( (1, -2) \)
6. \( (2, 1) \)
7. \( (3, 6) \)

### Graphing

The graph of the equation \( y = x^2 - 3 \) is a standard parabola that opens upwards. The vertex of this parabola can be found by rewriting the equation in vertex form. The vertex is at the point \( (0, -3) \), which is the lowest point of the graph.

- The parabola is symmetric about the y-axis, meaning that the left and right sides are mirror images of each other.

You can plot the points above:

- \( (-3, 6) \)
- \( (-2, 1) \)
- \( (-1, -2) \)
- \( (0, -3) \)
- \( (1, -2) \)
- \( (2, 1) \)
- \( (3, 6) \)

### Conclusion

To visualize, you can sketch these points and draw a smooth curve through them to represent the parabola \( y = x^2 - 3 \). The curve will pass through the vertex at \( (0, -3) \) and rise upwards on both sides, matching the points you calculated.