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Let's graph the function [tex]\( y = (x + 4)^2 - 6 \)[/tex]. To do this, we need to determine 7 points with integer coordinates on the graph of this function. Here are the steps to find these points:
1. Identify the function structure:
The function [tex]\( y = (x + 4)^2 - 6 \)[/tex] is a quadratic function in vertex form. The general form [tex]\( y = (x - h)^2 + k \)[/tex] indicates a parabola that opens upwards if the coefficient of the squared term is positive, which it is in this case.
2. Select integer x-values:
We will select integer x-values within a specific range. Here, we'll choose x-values from [tex]\(-10\)[/tex] to [tex]\(-4\)[/tex] inclusively.
3. Calculate the corresponding y-values:
For each selected x-value, we'll substitute it into the function to find the corresponding y-coordinate.
- For [tex]\( x = -10 \)[/tex]:
[tex]\[ y = (-10 + 4)^2 - 6 = (-6)^2 - 6 = 36 - 6 = 30 \][/tex]
So, one point is [tex]\((-10, 30)\)[/tex].
- For [tex]\( x = -9 \)[/tex]:
[tex]\[ y = (-9 + 4)^2 - 6 = (-5)^2 - 6 = 25 - 6 = 19 \][/tex]
So, another point is [tex]\((-9, 19)\)[/tex].
- For [tex]\( x = -8 \)[/tex]:
[tex]\[ y = (-8 + 4)^2 - 6 = (-4)^2 - 6 = 16 - 6 = 10 \][/tex]
So, another point is [tex]\((-8, 10)\)[/tex].
- For [tex]\( x = -7 \)[/tex]:
[tex]\[ y = (-7 + 4)^2 - 6 = (-3)^2 - 6 = 9 - 6 = 3 \][/tex]
So, another point is [tex]\((-7, 3)\)[/tex].
- For [tex]\( x = -6 \)[/tex]:
[tex]\[ y = (-6 + 4)^2 - 6 = (-2)^2 - 6 = 4 - 6 = -2 \][/tex]
So, another point is [tex]\((-6, -2)\)[/tex].
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = (-5 + 4)^2 - 6 = (-1)^2 - 6 = 1 - 6 = -5 \][/tex]
So, another point is [tex]\((-5, -5)\)[/tex].
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = (-4 + 4)^2 - 6 = (0)^2 - 6 = 0 - 6 = -6 \][/tex]
So, another point is [tex]\((-4, -6)\)[/tex].
4. Summary of the points:
The 7 points with integer coordinates that we have calculated are:
[tex]\[ (-10, 30), (-9, 19), (-8, 10), (-7, 3), (-6, -2), (-5, -5), (-4, -6) \][/tex]
With these points, you can plot the function [tex]\( y = (x + 4)^2 - 6 \)[/tex] on a coordinate plane. Simply plot each of these points and then draw a smooth parabolic curve passing through them to visualize the function.
1. Identify the function structure:
The function [tex]\( y = (x + 4)^2 - 6 \)[/tex] is a quadratic function in vertex form. The general form [tex]\( y = (x - h)^2 + k \)[/tex] indicates a parabola that opens upwards if the coefficient of the squared term is positive, which it is in this case.
2. Select integer x-values:
We will select integer x-values within a specific range. Here, we'll choose x-values from [tex]\(-10\)[/tex] to [tex]\(-4\)[/tex] inclusively.
3. Calculate the corresponding y-values:
For each selected x-value, we'll substitute it into the function to find the corresponding y-coordinate.
- For [tex]\( x = -10 \)[/tex]:
[tex]\[ y = (-10 + 4)^2 - 6 = (-6)^2 - 6 = 36 - 6 = 30 \][/tex]
So, one point is [tex]\((-10, 30)\)[/tex].
- For [tex]\( x = -9 \)[/tex]:
[tex]\[ y = (-9 + 4)^2 - 6 = (-5)^2 - 6 = 25 - 6 = 19 \][/tex]
So, another point is [tex]\((-9, 19)\)[/tex].
- For [tex]\( x = -8 \)[/tex]:
[tex]\[ y = (-8 + 4)^2 - 6 = (-4)^2 - 6 = 16 - 6 = 10 \][/tex]
So, another point is [tex]\((-8, 10)\)[/tex].
- For [tex]\( x = -7 \)[/tex]:
[tex]\[ y = (-7 + 4)^2 - 6 = (-3)^2 - 6 = 9 - 6 = 3 \][/tex]
So, another point is [tex]\((-7, 3)\)[/tex].
- For [tex]\( x = -6 \)[/tex]:
[tex]\[ y = (-6 + 4)^2 - 6 = (-2)^2 - 6 = 4 - 6 = -2 \][/tex]
So, another point is [tex]\((-6, -2)\)[/tex].
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = (-5 + 4)^2 - 6 = (-1)^2 - 6 = 1 - 6 = -5 \][/tex]
So, another point is [tex]\((-5, -5)\)[/tex].
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = (-4 + 4)^2 - 6 = (0)^2 - 6 = 0 - 6 = -6 \][/tex]
So, another point is [tex]\((-4, -6)\)[/tex].
4. Summary of the points:
The 7 points with integer coordinates that we have calculated are:
[tex]\[ (-10, 30), (-9, 19), (-8, 10), (-7, 3), (-6, -2), (-5, -5), (-4, -6) \][/tex]
With these points, you can plot the function [tex]\( y = (x + 4)^2 - 6 \)[/tex] on a coordinate plane. Simply plot each of these points and then draw a smooth parabolic curve passing through them to visualize the function.
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