Final answer:
To determine if a tangent line to the graph of f(x) = x² - 4 is parallel to the line 6x - 3y = 2, the slopes must match.
Step-by-step explanation:
Understanding Tangent Lines and Parallelism
To find the point on the graph of f(x) = x² - 4 where the tangent is parallel to the given line 6x - 3y = 2, we first need to find the slope of the line. The given line is in the standard form, which can be rewritten in the slope-intercept form as y = 2x - ⅓/3, giving us a slope of 2. A tangent line will be parallel to this line if it has the same slope.
The slope of the tangent to the graph of f(x) at any point is given by the derivative f'(x), which for our function is f'(x) = 2x. Setting this equal to 2 (the slope of the given line), we find that x = 1. However, a point with x = 1 does not match either of the points given in the question, which are (-2, 0) and (2, 0). The points provided may be on the graph of f(x), but the tangent lines at those points will have slopes of f'(-2) = -4 and f'(2) = 4, neither of which is equal to 2. Therefore, we can conclude that both statements are false.