Which are characteristics of the graph of the function f(x) = (x + 1)2 + 2? Check all that apply.

The domain is all real numbers.
The range is all real numbers greater than or equal to 1.
The y-intercept is 3.
The graph of the function is 1 unit up and 2 units to the left from the graph of y = x2.
The graph has two x-intercepts.

1 answer

Let's analyze the function \( f(x) = (x + 1)^2 + 2 \) step-by-step to determine which characteristics apply:

  1. Domain:

    • The expression \( (x + 1)^2 \) is defined for all real numbers \( x \). Therefore, the domain of \( f(x) \) is all real numbers.
  2. Range:

    • The term \( (x + 1)^2 \) reaches a minimum value of 0 (when \( x = -1 \)). Thus, \( f(x) \) achieves a minimum value of \( 0 + 2 = 2 \). Therefore, the range is all real numbers greater than or equal to 2, not 1.
  3. Y-intercept:

    • To find the y-intercept, we evaluate \( f(0) \): \[ f(0) = (0 + 1)^2 + 2 = 1^2 + 2 = 1 + 2 = 3. \]
    • So the y-intercept is indeed 3.
  4. Transformation from the graph of \( y = x^2 \):

    • The function \( f(x) = (x + 1)^2 + 2 \) represents a transformation of \( y = x^2 \):
      • The graph shifts 1 unit to the left (due to \( x + 1 \)) and 2 units up (due to \( +2 \)).
    • This statement is correct.
  5. X-intercepts:

    • To find the x-intercepts, we set \( f(x) = 0 \): \[ (x + 1)^2 + 2 = 0. \] However, since \( (x + 1)^2 \geq 0 \), \( (x + 1)^2 + 2 \geq 2 \), which means there are no x-intercepts.
    • Therefore, the graph does not have two x-intercepts.

Based on this analysis, the characteristics that apply to the function \( f(x) = (x + 1)^2 + 2 \) are:

  • The domain is all real numbers.
  • The y-intercept is 3.
  • The graph of the function is 1 unit up and 2 units to the left from the graph of \( y = x^2 \).

Thus, the correct answers are:

  • The domain is all real numbers.
  • The y-intercept is 3.
  • The graph of the function is 1 unit up and 2 units to the left from the graph of \( y = x^2 \).