Final answer:
The correct characteristics of the function f(x) = (x + 1)^2 + 2 include that the domain is all real numbers, and the y-intercept is 3. The range is all real numbers greater than or equal to 2, the graph is shifted 2 units up and 1 unit left compared to y = x^2, and the function does not have any real x-intercepts because the vertex is above the x-axis.
Step-by-step explanation:
The function f(x) = (x + 1)2 + 2 is a quadratic equation, which is a parabola when graphed. Let's analyze the given characteristics one by one:
- A) The domain of a quadratic function is all real numbers since there are no restrictions on the values that x can take.
- B) The range of the function is all real numbers greater than or equal to 2, not 1, because the vertex of the parabola is shifted 1 unit to the left and 2 units up due to the transformation applied to x2.
- C) To find the y-intercept, set x to 0. f(0) = (0 + 1)2 + 2 = 1 + 2 = 3. Therefore, the y-intercept is indeed 3.
- D) The graph is not 1 unit up and 2 units to the left from the graph of y = x2; it is actually 2 units up and 1 unit to the left.
- E) Quadratic functions may have two, one, or no real x-intercepts depending on the values of their coefficients. However, since the vertex is above the x-axis and the parabola opens upwards, this particular quadratic function will not have any real x-intercepts, as it does not cross the x-axis.
Considering the options given, we can determine that the correct characteristics of the graph of the function f(x) = (x + 1)2 + 2 are:
- The domain is all real numbers.
- The y-intercept is 3.1