Let's analyze the given function [tex]\( f(x) = (x+1)^2 + 2 \)[/tex] to determine its characteristics:
1. Domain:
The function [tex]\((x+1)^2 + 2\)[/tex] is defined for any real number [tex]\(x\)[/tex] because a quadratic function (squared term) and addition are operations that can be applied to all real numbers. Therefore, the domain is:
[tex]\[
\text{All real numbers}
\][/tex]
2. Range:
To determine the range, we should look at the minimum value of the function [tex]\( (x+1)^2 + 2 \)[/tex]. Since [tex]\( (x+1)^2 \)[/tex] is always non-negative (i.e., [tex]\(\geq 0\)[/tex]), the minimum value of [tex]\( f(x) \)[/tex] occurs when [tex]\( (x+1)^2 = 0 \)[/tex], which leads to:
[tex]\[
f(x) = 0 + 2 = 2
\][/tex]
Therefore, the range of the function is:
[tex]\[
\text{All real numbers greater than or equal to 2}
\][/tex]
3. [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept is the point where the function intersects the [tex]\( y \)[/tex]-axis, which is when [tex]\( x = 0 \)[/tex]. Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = (0 + 1)^2 + 2 = 1 + 2 = 3
\][/tex]
Thus, the [tex]\( y \)[/tex]-intercept is:
[tex]\[
3
\][/tex]
4. Graph Transformation:
The function can be written in the form [tex]\( f(x) = (x - h)^2 + k \)[/tex], which represents a parabola with a vertex at [tex]\( (h, k) \)[/tex]. Here, [tex]\( h = -1 \)[/tex] and [tex]\( k = 2 \)[/tex]. This means the graph of the function [tex]\( f(x) = x^2 \)[/tex] is shifted:
[tex]\[
\text{1 unit to the left} \quad \text{and} \quad \text{2 units up}
\][/tex]
5. [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts are the points where the function intersects the [tex]\( x \)[/tex]-axis, which means [tex]\( f(x) = 0 \)[/tex]. Solving for [tex]\( x \)[/tex]:
[tex]\[
(x+1)^2 + 2 = 0
\][/tex]
[tex]\[
(x+1)^2 = -2
\][/tex]
Since [tex]\( (x+1)^2 \)[/tex] is always non-negative, it cannot equal [tex]\(-2\)[/tex], which is a negative number. Hence, there are:
[tex]\[
\text{0 } x\text{-intercepts}
\][/tex]
Based on this analysis, the correct characteristics of the graph of the function [tex]\( f(x) = (x+1)^2 + 2 \)[/tex] are:
- The domain is all real numbers.
- The range is all real numbers greater than or equal to 2.
- The [tex]\( y \)[/tex]-intercept is 3.
- The graph of the function is 1 unit to the left and 2 units up from the graph of [tex]\( y = x^2 \)[/tex].
- The graph has no [tex]\( x \)[/tex]-intercepts.
