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].
