1. \( y=x^{3}+8 \)

It looks like you’re working with the function   y = x³ + 8. Since there isn’t a specific question attached to the function, here are several useful pieces of information and ways to work with it: 1. Factorization (using the sum‐of‐cubes formula):  The sum‐of‐cubes formula states that   a³ + b³ = (a + b)(a² – ab + b²).  Here, we can write 8 as 2³. Taking a = x and b = 2, we have   x³ + 8 = x³ + 2³ = (x + 2)(x² – 2x + 4). 2. Zeros of the function:  To find the real zero, set y = 0:   x³ + 8 = 0 ⟹ x³ = –8 ⟹ x = –2.  Thus, the function has a single real root, x = –2. (The quadratic factor, x² – 2x + 4, has a discriminant of (–2)² – 4·1·4 = 4 – 16 = –12, indicating two complex conjugate roots.) 3. Intercepts:  • y-intercept: Set x = 0:   y = 0³ + 8 = 8.   So the graph passes through (0, 8).  • x-intercept: As found above, the only x-intercept is (–2, 0). 4. End Behavior:  Since the function is a cubic (with positive leading coefficient):   • As x → +∞, y → +∞.   • As x → –∞, y → –∞. 5. Graph and Transformation:  The graph of y = x³ + 8 is the standard cubic function y = x³ shifted upward by 8 units. (Be careful: this is not a horizontal shift.) 6. Inflection Point:  For the standard cubic function y = x³, the inflection point is at (0, 0). An upward translation by 8 moves the inflection point to (0, 8). In summary, the function y = x³ + 8 can be rewritten as   y = (x + 2)(x² – 2x + 4), has a real zero at x = –2, a y-intercept at (0, 8), and its graph is a vertically translated version of y = x³.