Question

The function h(x) = (x – 3)3 + 1 is graphed on a coordinate grid. Which statements about the graph are true? Check all that apply.

The domain of the function is all real numbers.
The range of the function is all real numbers.
The graph is a translation 3 left and 1 up from the function f(x) = x3.
The graph crosses the y-axis once.
The graph crosses the x-axis three times.
The graph increases from negative infinity to positive infinity.

Answer 1

True:
- The domain is all real numbers. (Polynomials are defined for every real x.)
- The range is all real numbers. (A cubic with positive leading coefficient takes all real values.)
- The graph crosses the y-axis once. (h(0) = (0−3)^3 + 1 = −27 + 1 = −26, so one intersection at (0, −26).)
- The graph increases from negative infinity to positive infinity. (x^3 is strictly increasing and translations preserve that.)

False:
- "Translation 3 left and 1 up" is incorrect — h(x) = (x−3)^3 + 1 is f(x−3)+1, so it is shifted 3 units right and 1 unit up from f(x)=x^3.
- "Graph crosses the x-axis three times" is incorrect — solve (x−3)^3 + 1 = 0 ⇒ x = 2 is the only real root, so it crosses the x-axis once.