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To graph the equation [tex]\( y = 1000x^2 - 1800x \)[/tex] on the specified viewing window [tex]\([-5, 5, 1]\)[/tex] by [tex]\([-1000, 2000, 500]\)[/tex], follow these detailed steps:
1. Identify the equation and the viewing window:
- Equation: [tex]\( y = 1000x^2 - 1800x \)[/tex]
- x-axis range: [tex]\([-5, 5]\)[/tex] with tick marks every 1 unit
- y-axis range: [tex]\([-1000, 2000]\)[/tex] with tick marks every 500 units
2. Create a table of values:
Calculate the value of y for several values of x within the range [tex]\([-5, 5]\)[/tex] to ensure we have enough points to plot the curve accurately.
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = 1000(-5)^2 - 1800(-5) = 1000 \cdot 25 + 9000 = 25000 + 9000 = 34000 \][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = 1000(-4)^2 - 1800(-4) = 1000 \cdot 16 + 7200 = 16000 + 7200 = 23200 \][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = 1000(-3)^2 - 1800(-3) = 1000 \cdot 9 + 5400 = 9000 + 5400 = 14400 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = 1000(-2)^2 - 1800(-2) = 1000 \cdot 4 + 3600 = 4000 + 3600 = 7600 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 1000(-1)^2 - 1800(-1) = 1000 \cdot 1 + 1800 = 1000 + 1800 = 2800 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 1000(0)^2 - 1800(0) = 0 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 1000(1)^2 - 1800(1) = 1000 - 1800 = -800 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 1000(2)^2 - 1800(2) = 1000 \cdot 4 - 3600 = 4000 - 3600 = 400 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 1000(3)^2 - 1800(3) = 1000 \cdot 9 - 5400 = 9000 - 5400 = 3600 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 1000(4)^2 - 1800(4) = 1000 \cdot 16 - 7200 = 16000 - 7200 = 8800 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 1000(5)^2 - 1800(5) = 1000 \cdot 25 - 9000 = 25000 - 9000 = 16000 \][/tex]
3. Plot the points:
Use the calculated values to sketch the graph by plotting the points on the viewing window:
- [tex]\((-5, 34000)\)[/tex]
- [tex]\((-4, 23200)\)[/tex]
- [tex]\((-3, 14400)\)[/tex]
- [tex]\((-2, 7600)\)[/tex]
- [tex]\((-1, 2800)\)[/tex]
- [tex]\((0, 0)\)[/tex]
- [tex]\((1, -800)\)[/tex]
- [tex]\((2, 400)\)[/tex]
- [tex]\((3, 3600)\)[/tex]
- [tex]\((4, 8800)\)[/tex]
- [tex]\((5, 16000)\)[/tex]
4. Draw the curve:
Connect the points smoothly to form a parabolic curve. The vertex of this parabola, where it changes direction, is located at the minimum point since the coefficient of [tex]\(x^2\)[/tex] is positive. This happens at [tex]\(x = 0\)[/tex] where [tex]\(y = 0\)[/tex].
5. Adjust for the viewing window:
Given the range on the y-axis [tex]\([-1000, 2000]\)[/tex], the points calculated for larger values of [tex]\(x\)[/tex] (like [tex]\(\pm 4\)[/tex] or [tex]\(\pm 5\)[/tex]) are outside this range. Thus, focus on plotting the parts of the parabola that fall within [tex]\([-1000, 2000]\)[/tex] vertically.
6. Graph the equation:
You should see a parabola opening upwards with a vertex at [tex]\((0, 0)\)[/tex]. The sections of the curve within the y-axis from [tex]\([-1000, 2000]\)[/tex] should be well represented, especially around the minimum where [tex]\(x\)[/tex] is near 0.
Finally, ensure your graph has the correct scales on both axes: tick marks for every unit on the x-axis and every 500 units on the y-axis. This will provide a clear representation of the function [tex]\( y = 1000x^2 - 1800x \)[/tex] on the given window.
1. Identify the equation and the viewing window:
- Equation: [tex]\( y = 1000x^2 - 1800x \)[/tex]
- x-axis range: [tex]\([-5, 5]\)[/tex] with tick marks every 1 unit
- y-axis range: [tex]\([-1000, 2000]\)[/tex] with tick marks every 500 units
2. Create a table of values:
Calculate the value of y for several values of x within the range [tex]\([-5, 5]\)[/tex] to ensure we have enough points to plot the curve accurately.
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = 1000(-5)^2 - 1800(-5) = 1000 \cdot 25 + 9000 = 25000 + 9000 = 34000 \][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = 1000(-4)^2 - 1800(-4) = 1000 \cdot 16 + 7200 = 16000 + 7200 = 23200 \][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = 1000(-3)^2 - 1800(-3) = 1000 \cdot 9 + 5400 = 9000 + 5400 = 14400 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = 1000(-2)^2 - 1800(-2) = 1000 \cdot 4 + 3600 = 4000 + 3600 = 7600 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 1000(-1)^2 - 1800(-1) = 1000 \cdot 1 + 1800 = 1000 + 1800 = 2800 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 1000(0)^2 - 1800(0) = 0 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 1000(1)^2 - 1800(1) = 1000 - 1800 = -800 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 1000(2)^2 - 1800(2) = 1000 \cdot 4 - 3600 = 4000 - 3600 = 400 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 1000(3)^2 - 1800(3) = 1000 \cdot 9 - 5400 = 9000 - 5400 = 3600 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 1000(4)^2 - 1800(4) = 1000 \cdot 16 - 7200 = 16000 - 7200 = 8800 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 1000(5)^2 - 1800(5) = 1000 \cdot 25 - 9000 = 25000 - 9000 = 16000 \][/tex]
3. Plot the points:
Use the calculated values to sketch the graph by plotting the points on the viewing window:
- [tex]\((-5, 34000)\)[/tex]
- [tex]\((-4, 23200)\)[/tex]
- [tex]\((-3, 14400)\)[/tex]
- [tex]\((-2, 7600)\)[/tex]
- [tex]\((-1, 2800)\)[/tex]
- [tex]\((0, 0)\)[/tex]
- [tex]\((1, -800)\)[/tex]
- [tex]\((2, 400)\)[/tex]
- [tex]\((3, 3600)\)[/tex]
- [tex]\((4, 8800)\)[/tex]
- [tex]\((5, 16000)\)[/tex]
4. Draw the curve:
Connect the points smoothly to form a parabolic curve. The vertex of this parabola, where it changes direction, is located at the minimum point since the coefficient of [tex]\(x^2\)[/tex] is positive. This happens at [tex]\(x = 0\)[/tex] where [tex]\(y = 0\)[/tex].
5. Adjust for the viewing window:
Given the range on the y-axis [tex]\([-1000, 2000]\)[/tex], the points calculated for larger values of [tex]\(x\)[/tex] (like [tex]\(\pm 4\)[/tex] or [tex]\(\pm 5\)[/tex]) are outside this range. Thus, focus on plotting the parts of the parabola that fall within [tex]\([-1000, 2000]\)[/tex] vertically.
6. Graph the equation:
You should see a parabola opening upwards with a vertex at [tex]\((0, 0)\)[/tex]. The sections of the curve within the y-axis from [tex]\([-1000, 2000]\)[/tex] should be well represented, especially around the minimum where [tex]\(x\)[/tex] is near 0.
Finally, ensure your graph has the correct scales on both axes: tick marks for every unit on the x-axis and every 500 units on the y-axis. This will provide a clear representation of the function [tex]\( y = 1000x^2 - 1800x \)[/tex] on the given window.
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