Question

Vertex form with a range of (3, [infinity]) and vertical compression.

A) y=a(x−h)²+k
B) y=a(x−h)²−k
C) y=−a(x−h)² +k
D) y=−a(x−h)² −k

Answer 1

The correct equation is option C: y=-a(x-h)²+k. This represents an upside-down, vertically compressed parabola with its vertex at (h, k), ensuring the range starts at 3 and extends to infinity.

Step-by-step explanation:

The student has asked which vertex form equation has a range of (3, infinity) and includes a vertical compression. Given that the minimum value for the range is 3 and the function is supposed to have a vertical compression, we must focus on an equation that represents an upside-down parabola. The correct answer is option C: y=-a(x-h)²+k, because a negative 'a' value indicates an upside-down parabola (which ensures a maximum vertex value and an infinite range extending downwards), and the presence of 'k' adds a vertical shift to position the vertex at (h,3).

The 'k' value is directly correlated with the range's starting point since it determines the y-coordinate of the vertex. The 'a' value being negative is essential for ensuring that the parabola opens downward, while 'a' also being less than 1 (in absolute value) indicates that the parabola is vertically compressed.