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The image shows four lines representing direct variations, each passing through the origin (0,0). A direct variation is a linear relationship of the form \( y = kx \), where \( k \) is the coefficient or slope.

### Determining the sign and equation for each line:

1. **Line (1)** (Orange):
   - Slope \( k \): This line has a negative slope since it is decreasing from left to right.
   - From the graph, the slope \( k \) can be calculated by picking two points on the line, such as \((-3, 1)\) and \((3, -1)\):
   \[
   k = \frac{-1 - 1}{3 - (-3)} = \frac{-2}{6} = -\frac{1}{3}
   \]
   - Equation: \( y = -\frac{1}{3}x \)

2. **Line (2)** (Green):
   - Slope \( k \): This line has a negative slope since it is decreasing from left to right.
   - Picking points like \((-3, 3)\) and \((3, -3)\):
   \[
   k = \frac{-3 - 3}{3 - (-3)} = \frac{-6}{6} = -1
   \]
   - Equation: \( y = -x \)

3. **Line (3)** (Blue):
   - Slope \( k \): This line has a positive slope as it is increasing from left to right.
   - Picking points like \((2, 4)\) and \((4, 8)\):
   \[
   k = \frac{8 - 4}{4 - 2} = \frac{4}{2} = 2
   \]
   - Equation: \( y = 2x \)

4. **Line (4)** (Yellow):
   - Slope \( k \): This line has a positive slope as it is increasing from left to right.
   - Picking points like \((2, 1)\) and \((4, 2)\):
   \[
   k = \frac{2 - 1}{4 - 2} = \frac{1}{2}
   \]
   - Equation: \( y = \frac{1}{2}x \)

### Summary of Equations and Signs:
1. Line (1): \( y = -\frac{1}{3}x \), \( k < 0 \)
2. Line (2): \( y = -x \), \( k < 0 \)
3. Line (3): \( y = 2x \), \( k > 0 \)
4. Line (4): \( y = \frac{1}{2}x \), \( k > 0 \)

Do you want more details or have any questions?

### Related Questions:
1. How can you determine the slope \( k \) from a graph?
2. What is the significance of the sign of \( k \) in a direct variation?
3. How do you find the equation of a line given two points?
4. How would the graph change if the value of \( k \) increases?
5. What is the relationship between the steepness of a line and its slope?

**Tip:** Remember, the slope \( k \) in \( y = kx \) represents the rate of change of \( y \) with respect to \( x \). The greater the absolute value of \( k \), the steeper the line.

The picture shows graphs of four direct variations. Determine the sign of the coefficient k for each direct variation. Write the equation for each line.

Analyze graphs of four direct variations and determine the sign of the coefficient k. Learn how to calculate the slope using two points and write the equation for each line based on the direct variation formula y = kx. Suitable for students in Grades 8-10.

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