To determine the graph of the function \( f(x) = x^2 + 2x + 3 \), we should identify key features such as the vertex, direction in which the graph opens, and any additional important points.
1. Identify the Type of Function:
- The given function is a quadratic function because it has the form \( f(x) = ax^2 + bx + c \), with \( a = 1 \), \( b = 2 \), and \( c = 3 \).
2. Direction of the Parabola:
- Since the coefficient of \( x^2 \) (which is \( a \)) is positive (\( a = 1 \)), the parabola opens upwards.
3. Finding the Vertex:
- The vertex form of a quadratic function \( ax^2 + bx + c \) is given by completing the square.
- Start with \( f(x) = x^2 + 2x + 3 \).
- To complete the square, take the coefficient of \( x \) (which is 2), halve it (giving 1), and square it (giving 1). Add and subtract this square inside the function:
[tex]\[
f(x) = x^2 + 2x + 1 + 3 - 1
\][/tex]
- This results in:
[tex]\[
f(x) = (x + 1)^2 + 2
\][/tex]
- Therefore, the vertex form of the function is:
[tex]\[
f(x) = (x + 1)^2 + 2
\][/tex]
- From this form, it is clear that the vertex of the parabola is at \( (-1, 2) \).
4. Behavior of the Function:
- As \( x \) becomes very large in the positive or negative direction, the \( x^2 \) term dominates, causing \( f(x) \) to increase, confirming the parabola opens upwards.
5. Graphing the Function:
- The vertex at \( (-1, 2) \) is a critical point. Since the parabola opens upwards, it will be symmetric around a vertical line through the vertex (the axis of symmetry, \( x = -1 \)).
By these steps, we identified the key features of the graph of the function \( f(x) = x^2 + 2x + 3 \) and determined that it is a parabola opening upwards with its vertex at \( (-1, 2) \).
Thus, the correct graph will display these characteristics.
