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Bot GPT 3.5 To graph the parabola Y=(x+3)^2+1, we can start by finding the vertex, and then plot some additional points.
The vertex of the parabola can be found by taking the opposite of the x-coordinate term of the equation, which in this case is -3, as the x-coordinate of the vertex. The y-coordinate of the vertex is given by the constant term, which is +1. Therefore, the vertex is (-3, 1).
To find two points to the left of the vertex, we can substitute x-values that are less than -3 into the equation and solve for y. Let's take x = -4 and x = -5.
For x= -4:
Y = (-4 + 3)^2 + 1
= (-1)^2 + 1
= 1 + 1
= 2
So, the point (-4, 2) is on the parabola.
For x= -5:
Y = (-5 + 3)^2 + 1
= (-2)^2 + 1
= 4 + 1
= 5
So, the point (-5, 5) is also on the parabola.
To find two points to the right of the vertex, we can substitute x-values that are greater than -3 into the equation and solve for y. Let's take x = -2 and x = -1.
For x= -2:
Y = (-2 + 3)^2 + 1
= 1^2 + 1
= 1 + 1
= 2
So, the point (-2, 2) is on the parabola.
For x= -1:
Y = (-1 + 3)^2 + 1
= 2^2 + 1
= 4 + 1
= 5
So, the point (-1, 5) is also on the parabola.
Finally, we can also use the vertex to plot an additional point by taking the x-coordinate of the vertex and substituting it into the equation. Considering the vertex (-3, 1):
Y = (-3 + 3)^2 + 1
= 0^2 + 1
= 0 + 1
= 1
So, the point (-3, 1) is on the parabola.
Now, we can plot the points and graph the parabola:
Vertex: (-3, 1)
Two points to the left: (-4, 2) and (-5, 5)
Two points to the right: (-2, 2) and (-1, 5)
^
|
5 | ●
| .
|
4 |
|
|
3 |
| .
|
2 | .
|
|
1 | ●
|____________________________________
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
Note: The points are represented by ●, and the parabola is curved upwards, indicating it opens up.