What would the range be

Question

Determine the range of the function f(x) = 6x ^ 2 + 2x + 7

Answer 1

f(x) = 6x² + 2x + 7 = 6(x + 1/6)² + 41/6;

Domain: [41/6, ∞)

Answer 2

f(x) is an upward opening parabola. Range is from the vertex to positive infinity

6x^2 +2x + 7 = 0

divide by 6.

factor out 6 for the two x terms

6(x^2 + x/3) + 7 = 0 complete the square, take half of 1/3= 1/6 and square it to get 1/36, add and subtract 6/36

6(x^2 +x/3 + 1/36) + 7 -6/36

6(x+1/6)^2 + 7 - 1/6 = 0

6(x+1/6)^2 + 6 5/6 = 0 this is in vertex form a(x-h)^2 + k where (h,k) is the vertex and k is the minimum value of f(x)

vertex is (-1/6, 41/6)

range is the interval [41/6, + infinity)