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To graph the line with a slope of [tex]\(-\frac{1}{2}\)[/tex] passing through the point [tex]\((2, -4)\)[/tex], we need to follow these steps:
1. Identify the Slope and the Point: We are given the slope [tex]\(m = -\frac{1}{2}\)[/tex] and the point [tex]\((2, -4)\)[/tex].
2. Equation of the Line: Use the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope. Plugging in the given point [tex]\((2, -4)\)[/tex] and the slope [tex]\(-\frac{1}{2}\)[/tex]:
[tex]\[ y - (-4) = -\frac{1}{2}(x - 2) \][/tex]
Simplify this equation:
[tex]\[ y + 4 = -\frac{1}{2}(x - 2) \][/tex]
3. Rewriting the Equation: Distribute the slope on the right-hand side:
[tex]\[ y + 4 = -\frac{1}{2}x + 1 \][/tex]
Subtract 4 from both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{1}{2}x + 1 - 4 \][/tex]
[tex]\[ y = -\frac{1}{2}x - 3 \][/tex]
Now we have the slope-intercept form of the equation, [tex]\(y = -\frac{1}{2}x - 3\)[/tex].
4. Sample Points on [tex]\(x\)[/tex]-Axis: To plot the line, we can calculate [tex]\(y\)[/tex] for various [tex]\(x\)[/tex] values. Let's choose values around the given point [tex]\(x = 2\)[/tex]:
- For [tex]\(x = -8\)[/tex]:
[tex]\[ y = -\frac{1}{2}(-8) - 3 = 4 - 3 = 1 \][/tex]
- For [tex]\(x = -7.95\)[/tex]:
[tex]\[ y = -\frac{1}{2}(-7.95) - 3 \approx 3.975 - 3 = 0.975 \][/tex]
- For [tex]\(x = -7.9\)[/tex]:
[tex]\[ y = -\frac{1}{2}(-7.9) - 3 = 3.95 - 3 = 0.95 \][/tex]
And so on.
Here are some calculated points:
[tex]\[ \begin{align*} (-8, 1), \\ (-7.95, 0.975), \\ (-7.9, 0.95), \\ \vdots \\ (2, -4), \\ \vdots \\ (8, -7), \\ (12, -9) \end{align*} \][/tex]
5. Plotting the Points: Plotting these ordered pairs on the graph, we get:
- Start with the point [tex]\((2, -4)\)[/tex], which is our given point.
- Use the slope to find another point; since the slope is [tex]\(-\frac{1}{2}\)[/tex], for every unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by [tex]\(\frac{1}{2}\)[/tex].
6. Drawing the Line: Connect the points with a straight line. Since the slope is negative, the line will fall from left to right.
The graphic representation of the line passing through these calculated points gives the line:
[tex]\[ \begin{align*} \begin{array}{|c|c|} \hline \text{x} & \text{y} \\ \hline -8 & 1 \\ -7.95 & 0.975 \\ -7.9 & 0.95 \\ 2 & -4 \\ 8 & -7 \\ 12 & -9 \\ \hline \end{array} \end{align*} \][/tex]
In conclusion, the plot of the line with a slope of [tex]\(-\frac{1}{2}\)[/tex] passing through [tex]\((2, -4)\)[/tex] extends through the points calculated, and it follows the linear equation [tex]\(y = -\frac{1}{2}x - 3\)[/tex].
1. Identify the Slope and the Point: We are given the slope [tex]\(m = -\frac{1}{2}\)[/tex] and the point [tex]\((2, -4)\)[/tex].
2. Equation of the Line: Use the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope. Plugging in the given point [tex]\((2, -4)\)[/tex] and the slope [tex]\(-\frac{1}{2}\)[/tex]:
[tex]\[ y - (-4) = -\frac{1}{2}(x - 2) \][/tex]
Simplify this equation:
[tex]\[ y + 4 = -\frac{1}{2}(x - 2) \][/tex]
3. Rewriting the Equation: Distribute the slope on the right-hand side:
[tex]\[ y + 4 = -\frac{1}{2}x + 1 \][/tex]
Subtract 4 from both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{1}{2}x + 1 - 4 \][/tex]
[tex]\[ y = -\frac{1}{2}x - 3 \][/tex]
Now we have the slope-intercept form of the equation, [tex]\(y = -\frac{1}{2}x - 3\)[/tex].
4. Sample Points on [tex]\(x\)[/tex]-Axis: To plot the line, we can calculate [tex]\(y\)[/tex] for various [tex]\(x\)[/tex] values. Let's choose values around the given point [tex]\(x = 2\)[/tex]:
- For [tex]\(x = -8\)[/tex]:
[tex]\[ y = -\frac{1}{2}(-8) - 3 = 4 - 3 = 1 \][/tex]
- For [tex]\(x = -7.95\)[/tex]:
[tex]\[ y = -\frac{1}{2}(-7.95) - 3 \approx 3.975 - 3 = 0.975 \][/tex]
- For [tex]\(x = -7.9\)[/tex]:
[tex]\[ y = -\frac{1}{2}(-7.9) - 3 = 3.95 - 3 = 0.95 \][/tex]
And so on.
Here are some calculated points:
[tex]\[ \begin{align*} (-8, 1), \\ (-7.95, 0.975), \\ (-7.9, 0.95), \\ \vdots \\ (2, -4), \\ \vdots \\ (8, -7), \\ (12, -9) \end{align*} \][/tex]
5. Plotting the Points: Plotting these ordered pairs on the graph, we get:
- Start with the point [tex]\((2, -4)\)[/tex], which is our given point.
- Use the slope to find another point; since the slope is [tex]\(-\frac{1}{2}\)[/tex], for every unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by [tex]\(\frac{1}{2}\)[/tex].
6. Drawing the Line: Connect the points with a straight line. Since the slope is negative, the line will fall from left to right.
The graphic representation of the line passing through these calculated points gives the line:
[tex]\[ \begin{align*} \begin{array}{|c|c|} \hline \text{x} & \text{y} \\ \hline -8 & 1 \\ -7.95 & 0.975 \\ -7.9 & 0.95 \\ 2 & -4 \\ 8 & -7 \\ 12 & -9 \\ \hline \end{array} \end{align*} \][/tex]
In conclusion, the plot of the line with a slope of [tex]\(-\frac{1}{2}\)[/tex] passing through [tex]\((2, -4)\)[/tex] extends through the points calculated, and it follows the linear equation [tex]\(y = -\frac{1}{2}x - 3\)[/tex].
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