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

Question

Answer should be a whole number or simplified fraction
Solid line upward v shape (0,0)(-1,3),;1,3),(-2,6),(2,6)
Dotted line points (9,0),(-1,1),(1,1),(-2,2),(2,2), (-3,3),(3,3)

Answer 1

To determine the factor \( k \) that would cause the vertical compression from the dotted line (the dilated function) to the solid line (the original function), we can compare corresponding points from both functions.

Let's pick some points from both functions:

1. The pointed vertical shape (solid line) has the points:

   • (0, 0)
   • (-1, 3)
   • (1, 3)
   • (-2, 6)
   • (2, 6)

2. The dilated function (dotted line) has the points:

   • (9, 0)
   • (-1, 1)
   • (1, 1)
   • (-2, 2)
   • (2, 2)
   • (-3, 3)
   • (3, 3)

Next, we can compare corresponding points to find \( k \). A key pair of points to consider is (-1, 3) from the solid line and (-1, 1) from the dotted line.

To find \( k \), we relate the y-values of these points:

\[ \text{Original Function (Solid)}: y = 3 \quad \text{at } x = -1 \] \[ \text{Dilated Function (Dotted)}: y = 1 \quad \text{at } x = -1 \]

Now we set up the vertical compression equation: \[ y_{\text{dilated}} = k \cdot y_{\text{original}} \] Substituting the values we have: \[ 1 = k \cdot 3 \] Now, solving for \( k \): \[ k = \frac{1}{3} \]

Thus, the value of \( k \) that would cause the vertical compression from the dilated function to the solid line is

\[ \boxed{\frac{1}{3}} \]

Answer 2

Correction on dotted line (0,0)

Answer 3

Thank you for the correction regarding the points. Let's reevaluate the scenario with the updated information on the dotted line, which has the point (0, 0) among other points.

The points you provided now for the solid line (vertical “V” shape) and dotted line are:

Solid Line (original function):

• (0, 0)
• (-1, 3)
• (1, 3)
• (-2, 6)
• (2, 6)

Dotted Line (dilated function):

• (0, 0) (the same point)
• (-1, 1)
• (1, 1)
• (-2, 2)
• (2, 2)
• (-3, 3)
• (3, 3)

Since (0, 0) is a common point, we can choose one of the other pairs of points to find \( k \). Let's take the points (-1, 3) from the solid line and (-1, 1) from the dotted line.

Using these points, we have:

Original Function (Solid): \[ y_{\text{solid}} = 3 \quad \text{at } x = -1 \]

Dilated Function (Dotted): \[ y_{\text{dotted}} = 1 \quad \text{at } x = -1 \]

The relationship for vertical compression is: \[ y_{\text{dotted}} = k \cdot y_{\text{solid}} \] Substituting the values we have: \[ 1 = k \cdot 3 \]

Now, solving for \( k \): \[ k = \frac{1}{3} \]

Thus, the value of \( k \) that would cause the vertical compression from the dilated function to the solid line is:

\[ \boxed{\frac{1}{3}} \]