The vertex of a graph typically refers to the highest or lowest point of a parabola, which is not applicable for cubic functions like the one described in the function f(x) = x³ + 7.
Instead of a vertex, cubic functions have an inflection point where the curve changes direction. The function you provided, f(x) = x³ + 7, is a cubic polynomial where the term x³ dominates the behavior at extreme values of x.
However, we can determine the critical point of the function: to find where the function changes from increasing to decreasing (or vice versa), we can calculate the derivative, f'(x). The derivative of the function is:
f'(x) = 3x²
Setting the derivative equal to zero to find critical points gives:
3x² = 0
This results in:
x = 0
Now, substituting x = 0 back into the original function to find the corresponding y value:
f(0) = 0³ + 7 = 7
Therefore, the point (0, 7) acts as the inflection point of the graph. The graph of the function increases without bound on both sides and has a distinctive S-shape due to the odd degree of the polynomial.
In summary, while the function f(x) = x³ + 7 does not have a vertex in the traditional sense, its inflection point is at (0, 7), indicating a change in the curve’s direction.