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To find seven ordered pairs [tex]\((x, y)\)[/tex] that satisfy the equation [tex]\( y = 6 - x^2 \)[/tex], we will select seven different values for [tex]\( x \)[/tex] and then calculate the corresponding values of [tex]\( y \)[/tex].
Let's choose the following values for [tex]\( x \)[/tex]: [tex]\(-3, -2, -1, 0, 1, 2, 3\)[/tex].
Now, we'll substitute each [tex]\( x \)[/tex] value into the equation and solve for [tex]\( y \)[/tex].
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = 6 - (-3)^2 = 6 - 9 = -3 \][/tex]
The ordered pair is [tex]\((-3, -3)\)[/tex].
2. For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = 6 - (-2)^2 = 6 - 4 = 2 \][/tex]
The ordered pair is [tex]\((-2, 2)\)[/tex].
3. For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 6 - (-1)^2 = 6 - 1 = 5 \][/tex]
The ordered pair is [tex]\((-1, 5)\)[/tex].
4. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 6 - 0^2 = 6 - 0 = 6 \][/tex]
The ordered pair is [tex]\((0, 6)\)[/tex].
5. For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 6 - 1^2 = 6 - 1 = 5 \][/tex]
The ordered pair is [tex]\((1, 5)\)[/tex].
6. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 6 - 2^2 = 6 - 4 = 2 \][/tex]
The ordered pair is [tex]\((2, 2)\)[/tex].
7. For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 6 - 3^2 = 6 - 9 = -3 \][/tex]
The ordered pair is [tex]\((3, -3)\)[/tex].
So, the seven ordered pairs that satisfy the equation [tex]\( y = 6 - x^2 \)[/tex] are:
[tex]\[ (-3, -3), (-2, 2), (-1, 5), (0, 6), (1, 5), (2, 2), (3, -3) \][/tex]
Now, let's determine the graph of the equation [tex]\( y = 6 - x^2 \)[/tex].
This equation represents a parabola that opens downwards because the coefficient of the [tex]\( x^2 \)[/tex] term is negative (-1). The vertex of the parabola is at [tex]\((0, 6)\)[/tex], the highest point on the graph, and the parabola is symmetric about the y-axis.
Using the ordered pairs we calculated, we can plot these points on a coordinate plane to visualize the shape of the graph. When we plot [tex]\((-3, -3), (-2, 2), (-1, 5), (0, 6), (1, 5), (2, 2), (3, -3)\)[/tex], we will see the parabolic curve extending from the vertex (0, 6) and opening downward through these points.
Let's choose the following values for [tex]\( x \)[/tex]: [tex]\(-3, -2, -1, 0, 1, 2, 3\)[/tex].
Now, we'll substitute each [tex]\( x \)[/tex] value into the equation and solve for [tex]\( y \)[/tex].
1. For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = 6 - (-3)^2 = 6 - 9 = -3 \][/tex]
The ordered pair is [tex]\((-3, -3)\)[/tex].
2. For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = 6 - (-2)^2 = 6 - 4 = 2 \][/tex]
The ordered pair is [tex]\((-2, 2)\)[/tex].
3. For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 6 - (-1)^2 = 6 - 1 = 5 \][/tex]
The ordered pair is [tex]\((-1, 5)\)[/tex].
4. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 6 - 0^2 = 6 - 0 = 6 \][/tex]
The ordered pair is [tex]\((0, 6)\)[/tex].
5. For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 6 - 1^2 = 6 - 1 = 5 \][/tex]
The ordered pair is [tex]\((1, 5)\)[/tex].
6. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 6 - 2^2 = 6 - 4 = 2 \][/tex]
The ordered pair is [tex]\((2, 2)\)[/tex].
7. For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 6 - 3^2 = 6 - 9 = -3 \][/tex]
The ordered pair is [tex]\((3, -3)\)[/tex].
So, the seven ordered pairs that satisfy the equation [tex]\( y = 6 - x^2 \)[/tex] are:
[tex]\[ (-3, -3), (-2, 2), (-1, 5), (0, 6), (1, 5), (2, 2), (3, -3) \][/tex]
Now, let's determine the graph of the equation [tex]\( y = 6 - x^2 \)[/tex].
This equation represents a parabola that opens downwards because the coefficient of the [tex]\( x^2 \)[/tex] term is negative (-1). The vertex of the parabola is at [tex]\((0, 6)\)[/tex], the highest point on the graph, and the parabola is symmetric about the y-axis.
Using the ordered pairs we calculated, we can plot these points on a coordinate plane to visualize the shape of the graph. When we plot [tex]\((-3, -3), (-2, 2), (-1, 5), (0, 6), (1, 5), (2, 2), (3, -3)\)[/tex], we will see the parabolic curve extending from the vertex (0, 6) and opening downward through these points.
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