Question
Find seven ordered pairs to the equation y equals 9 minus x squared. Then determine its graph.
Answers
GPT-4o mini
To find seven ordered pairs for the equation \( y = 9 - x^2 \), we can choose different values for \( x \) and compute the corresponding \( y \) values.
Let's choose the following values for \( x \): -3, -2, -1, 0, 1, 2, and 3.
Now, we will calculate \( y \) for each of these \( x \) values:
1. For \( x = -3 \):
\[
y = 9 - (-3)^2 = 9 - 9 = 0 \quad \Rightarrow \quad (-3, 0)
\]
2. For \( x = -2 \):
\[
y = 9 - (-2)^2 = 9 - 4 = 5 \quad \Rightarrow \quad (-2, 5)
\]
3. For \( x = -1 \):
\[
y = 9 - (-1)^2 = 9 - 1 = 8 \quad \Rightarrow \quad (-1, 8)
\]
4. For \( x = 0 \):
\[
y = 9 - 0^2 = 9 - 0 = 9 \quad \Rightarrow \quad (0, 9)
\]
5. For \( x = 1 \):
\[
y = 9 - 1^2 = 9 - 1 = 8 \quad \Rightarrow \quad (1, 8)
\]
6. For \( x = 2 \):
\[
y = 9 - 2^2 = 9 - 4 = 5 \quad \Rightarrow \quad (2, 5)
\]
7. For \( x = 3 \):
\[
y = 9 - 3^2 = 9 - 9 = 0 \quad \Rightarrow \quad (3, 0)
\]
Now, we have the seven ordered pairs:
1. \((-3, 0)\)
2. \((-2, 5)\)
3. \((-1, 8)\)
4. \((0, 9)\)
5. \((1, 8)\)
6. \((2, 5)\)
7. \((3, 0)\)
### Graphing the Equation
The graph of the equation \( y = 9 - x^2 \) is a downward-opening parabola. Here are some key characteristics:
- The vertex of the parabola is at the point \((0, 9)\).
- The parabola is symmetric about the y-axis, meaning that the points on the left (negative \( x \)) have identical \( y \) values to the corresponding points on the right (positive \( x \)).
- The parabola intersects the y-axis at \( (0, 9) \) and the x-axis at the points \((-3, 0)\) and \((3, 0)\).
You can plot the points on a coordinate plane and draw a smooth curve through them to illustrate the shape of the parabola. The shape will peak at the vertex and slope downwards on both sides.
Let's choose the following values for \( x \): -3, -2, -1, 0, 1, 2, and 3.
Now, we will calculate \( y \) for each of these \( x \) values:
1. For \( x = -3 \):
\[
y = 9 - (-3)^2 = 9 - 9 = 0 \quad \Rightarrow \quad (-3, 0)
\]
2. For \( x = -2 \):
\[
y = 9 - (-2)^2 = 9 - 4 = 5 \quad \Rightarrow \quad (-2, 5)
\]
3. For \( x = -1 \):
\[
y = 9 - (-1)^2 = 9 - 1 = 8 \quad \Rightarrow \quad (-1, 8)
\]
4. For \( x = 0 \):
\[
y = 9 - 0^2 = 9 - 0 = 9 \quad \Rightarrow \quad (0, 9)
\]
5. For \( x = 1 \):
\[
y = 9 - 1^2 = 9 - 1 = 8 \quad \Rightarrow \quad (1, 8)
\]
6. For \( x = 2 \):
\[
y = 9 - 2^2 = 9 - 4 = 5 \quad \Rightarrow \quad (2, 5)
\]
7. For \( x = 3 \):
\[
y = 9 - 3^2 = 9 - 9 = 0 \quad \Rightarrow \quad (3, 0)
\]
Now, we have the seven ordered pairs:
1. \((-3, 0)\)
2. \((-2, 5)\)
3. \((-1, 8)\)
4. \((0, 9)\)
5. \((1, 8)\)
6. \((2, 5)\)
7. \((3, 0)\)
### Graphing the Equation
The graph of the equation \( y = 9 - x^2 \) is a downward-opening parabola. Here are some key characteristics:
- The vertex of the parabola is at the point \((0, 9)\).
- The parabola is symmetric about the y-axis, meaning that the points on the left (negative \( x \)) have identical \( y \) values to the corresponding points on the right (positive \( x \)).
- The parabola intersects the y-axis at \( (0, 9) \) and the x-axis at the points \((-3, 0)\) and \((3, 0)\).
You can plot the points on a coordinate plane and draw a smooth curve through them to illustrate the shape of the parabola. The shape will peak at the vertex and slope downwards on both sides.