To determine the value of \( k \) for the vertical stretch from the original function (solid parabola) to the dilated function (dotted parabola), you would typically compare the two parabolas' equations.
Assuming the equations of the parabolas are in the form:
- Solid parabola (original): \( y = a(x - h)^2 + k \)
- Dotted parabola (dilated): \( y = b(x - h)^2 + k \)
Here:
- \( (h, k) \) is the vertex of both parabolas.
- \( a \) and \( b \) are the coefficients affecting the width and direction (opening downward for both, hence \( a < 0 \) and \( b < 0 \)).
- The value of \( k \) can be found by taking the ratio of the coefficients \( b \) and \( a \):
\[ k = \left| \frac{b}{a} \right| \]
In your case, without specific equations or values, I cannot provide a numerical answer. If you have those values, simply plug them into the equation above to compute \( k \). If \( a = -2 \) (for the solid parabola) and \( b = -4 \) (for the dotted parabola), for example, then:
\[ k = \left| \frac{-4}{-2} \right| = 2 \]
So if you provide the equations or specific coefficients, I can help you find the value of \( k \).