To graph the line with a slope of -1 passing through the point [tex]\((2, -2)\)[/tex], follow these steps:
### Step 1: Identify the Information Given
- Point through which the line passes: [tex]\((2, -2)\)[/tex]
- Slope ([tex]\(m\)[/tex]): -1
### Step 2: Use the Point-Slope Form to Find the Equation of the Line
The point-slope form of a line equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the given point and [tex]\(m\)[/tex] is the slope.
Substitute the given point [tex]\((2, -2)\)[/tex] and the slope [tex]\(-1\)[/tex] into this equation:
[tex]\[ y - (-2) = -1(x - 2) \][/tex]
Simplify this equation:
[tex]\[ y + 2 = -1(x - 2) \][/tex]
[tex]\[ y + 2 = -x + 2 \][/tex]
[tex]\[ y = -x + 2 - 2 \][/tex]
[tex]\[ y = -x \][/tex]
So, the equation of the line is:
[tex]\[ y = -x \][/tex]
### Step 3: Plot the Line Using the Equation
To plot the line [tex]\(y = -x\)[/tex], follow these steps:
1. Identify Points on the Line: Using the equation [tex]\(y = -x\)[/tex], select a few values for [tex]\(x\)[/tex] and solve for [tex]\(y\)[/tex]:
- If [tex]\(x = -2\)[/tex], then [tex]\(y = -(-2) = 2\)[/tex].
- If [tex]\(x = -1\)[/tex], then [tex]\(y = -(-1) = 1\)[/tex].
- If [tex]\(x = 0\)[/tex], then [tex]\(y = -(0) = 0\)[/tex].
- If [tex]\(x = 1\)[/tex], then [tex]\(y = -(1) = -1\)[/tex].
- If [tex]\(x = 2\)[/tex], then [tex]\(y = -(2) = -2\)[/tex].
2. Plot these points on a coordinate plane: [tex]\((-2, 2)\)[/tex], [tex]\((-1, 1)\)[/tex], [tex]\((0, 0)\)[/tex], [tex]\((1, -1)\)[/tex], [tex]\((2, -2)\)[/tex].
3. Draw the Line: Connect these points with a straight line.
### Step 4: Verify the Specific Point
Ensure that the point [tex]\((2, -2)\)[/tex] lies on the line. If we substitute [tex]\(x = 2\)[/tex] into the equation [tex]\(y = -x\)[/tex], we get:
[tex]\[ y = -2 \][/tex]
which matches the given point [tex]\((2, -2)\)[/tex].
### Step 5: Label the Axes and the Points
- Label the x-axis and y-axis with appropriate scale and units.
- Mark and label the point [tex]\((2, -2)\)[/tex] on the graph.
### Final Graph
After plotting the points and drawing the line through them, the line [tex]\(y = -x\)[/tex] with a slope of -1 passing through the point [tex]\((2, -2)\)[/tex] will be visibly represented on the coordinate plane. You will also see that the point [tex]\((2, -2)\)[/tex] lies on this line.
