Identify the factor k that would cause the vertical compression (solid line) to the dilated function (the dotted line). Note: It might be helpful to notice a point on the dilated function and find the corresponding point on the original function to help determine the value of k.

To identify the factor \( k \) that causes a vertical compression of a function, you can analyze the relationship between points on the original function and the dilated function.

1. Understand the relationship: A vertical compression by a factor of \( k \) means that if \( y = f(x) \) is the original function, then after compression, the new function can be expressed as \( y = \frac{1}{k} f(x) \).

2. Identify points: Check a specific point on the dilated function (dotted line) and find the corresponding point on the original function (the solid line). For example, if you have a point on the solid line \( (x_0, y_0) \) and a point on the dotted line \( (x_0, y_1) \), then you can express this as: \[ y_1 = \frac{1}{k} y_0 \]

3. Calculate \( k \): Rearranging the above equation gives: \[ k = \frac{y_0}{y_1} \]

4. Apply: Substitute the \( y \)-values of the corresponding points you identified.

5. Conclusion: Once you find the value of \( k \), you will know how much the original function has been compressed vertically to achieve the dilated function.

If you have specific points in mind or additional details related to the functions, please provide them, and I can help you compute \( k \).

the dotted line: vertex (0,0) ; points (-2,2) , (-1,1) , (1,1) , (2,2)
the solid line :vertex (0,0); points (-1,3) , (-2,6) , (1,3) , (2,6)

To find the vertical compression factor \( k \) between the dotted line (dilated function) and the solid line (original function), we need to compare corresponding points on each line, as you have provided.

Given points:

1. Dotted line (dilated function):

   • Vertex: \( (0, 0) \)
   • Points: \( (-2, 2) \), \( (-1, 1) \), \( (1, 1) \), \( (2, 2) \)

2. Solid line (original function):

   • Vertex: \( (0, 0) \)
   • Points: \( (-2, 6) \), \( (-1, 3) \), \( (1, 3) \), \( (2, 6) \)

Now, let's find \( k \) using corresponding points.

Using the point \((-2, 2)\) and \((-2, 6)\):

1. For the solid line point:
   \[ y_0 = 6 \]

2. For the dotted line point:
   \[ y_1 = 2 \]

Now, apply the formula: \[ k = \frac{y_0}{y_1} = \frac{6}{2} = 3 \]

Using another point \((-1, 1)\) and \((-1, 3)\):

1. For the solid line point:
   \[ y_0 = 3 \]

2. For the dotted line point:
   \[ y_1 = 1 \]

Now apply the formula again: \[ k = \frac{y_0}{y_1} = \frac{3}{1} = 3 \]

Using yet another point \((1, 1)\) and \((1, 3)\):

1. For the solid line point:
   \[ y_0 = 3 \]

2. For the dotted line point:
   \[ y_1 = 1 \]

Now apply the formula: \[ k = \frac{y_0}{y_1} = \frac{3}{1} = 3 \]

Using the last point \((2, 2)\) and \((2, 6)\):

1. For the solid line point:
   \[ y_0 = 6 \]

2. For the dotted line point:
   \[ y_1 = 2 \]

Now apply the formula: \[ k = \frac{y_0}{y_1} = \frac{6}{2} = 3 \]

Conclusion

In all cases, the vertical compression factor \( k \) is \( 3 \). Therefore, the solid line (original function) is compressed by a factor of \( k = 3 \) to produce the dotted line (dilated function).