Final answer:
To determine which of the given options is always correct when the graph of the quadratic function touches the x-axis, we analyze the discriminant. Option C (a^2 = 2c) is always correct because it satisfies the condition when only one real root is present.
Step-by-step explanation:
To determine which of the options is always correct when the graph of the given quadratic function touches the x-axis, we need to analyze its discriminant. When a quadratic touches the x-axis, it means it has only one real root. In this case, the discriminant (b^2 - 4ac) must be equal to zero. With a=3, b=6a, and c=6c, we can substitute these values into the discriminant equation and simplify to find the correct option. Evaluating the discriminant, we get (6a)^2 - 4(3)(6c) = 36a^2 - 72c.
Considering the options:
A) a^2 = 8c. Substituting a^2 = 8c into the discriminant equation, we get 36(8c) - 72c = 288c - 72c = 216c. So, option A is not always correct.
B) c = 4a^2. Substituting c = 4a^2 into the discriminant equation, we get 36a^2 - 72(4a^2) = 36a^2 - 288a^2 = -252a^2. So, option B is not always correct.
C) a^2 = 2c. Substituting a^2 = 2c into the discriminant equation, we get 36(2c) - 72c = 72c - 72c = 0. So, option C is always correct.
D) c^2 = 8a. There is no direct relation between c and a^2, so option D is not always correct.
Therefore, the correct option is C) a^2 = 2c.