To graph the line that represents the equation [tex]\( y + 2 = \frac{1}{2}(x + 2) \)[/tex], follow these steps:
### Step 1: Simplify the Equation
First, we need to rewrite the given equation in the slope-intercept form [tex]\( y = mx + c \)[/tex].
Given:
[tex]\[ y + 2 = \frac{1}{2}(x + 2) \][/tex]
Expand and simplify the equation:
[tex]\[ y + 2 = \frac{1}{2}x + \frac{1}{2}(2) \][/tex]
[tex]\[ y + 2 = \frac{1}{2}x + 1 \][/tex]
Subtract 2 from both sides:
[tex]\[ y = \frac{1}{2}x + 1 - 2 \][/tex]
[tex]\[ y = \frac{1}{2}x - 1 \][/tex]
So, the equation in slope-intercept form is:
[tex]\[ y = \frac{1}{2}x - 1 \][/tex]
### Step 2: Identify Key Points
To plot the line, identify a few key points by substituting specific [tex]\( x \)[/tex]-values into the equation to find the corresponding [tex]\( y \)[/tex]-values.
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = \frac{1}{2}(-2) - 1 \][/tex]
[tex]\[ y = -1 - 1 \][/tex]
[tex]\[ y = -2 \][/tex]
Point: [tex]\( (-2, -2) \)[/tex]
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{1}{2}(0) - 1 \][/tex]
[tex]\[ y = -1 \][/tex]
Point: [tex]\( (0, -1) \)[/tex]
3. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = \frac{1}{2}(2) - 1 \][/tex]
[tex]\[ y = 1 - 1 \][/tex]
[tex]\[ y = 0 \][/tex]
Point: [tex]\( (2, 0) \)[/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = \frac{1}{2}(4) - 1 \][/tex]
[tex]\[ y = 2 - 1 \][/tex]
[tex]\[ y = 1 \][/tex]
Point: [tex]\( (4, 1) \)[/tex]
### Step 3: Plot the Points
Plot the points on graph paper or a coordinate plane:
- [tex]\( (-2, -2) \)[/tex]
- [tex]\( (0, -1) \)[/tex]
- [tex]\( (2, 0) \)[/tex]
- [tex]\( (4, 1) \)[/tex]
### Step 4: Draw the Line
Draw a straight line through the points to represent the equation [tex]\( y = \frac{1}{2}x - 1 \)[/tex].
### Final Graph
Your final graph should show a line passing through the points [tex]\((-2, -2)\)[/tex], [tex]\((0, -1)\)[/tex], [tex]\((2, 0)\)[/tex], and [tex]\((4, 1)\)[/tex]. This line represents the equation [tex]\( y = \frac{1}{2}x - 1 \)[/tex].
