The function f(x) = x² + 2x + 3 is a quadratic function. To understand its graph, we can start by finding its vertex, axis of symmetry, and y-intercept.
1. **Finding the Vertex:**
The vertex of a quadratic function in the form f(x) = ax² + bx + c can be found using the formula:
x_vertex = -b / (2a)
In this case, a = 1 and b = 2, so:
x_vertex = -2 / (2 * 1) = -1
Now we plug this back into the function to get the y-coordinate of the vertex:
f(-1) = (-1)² + 2(-1) + 3 = 1 – 2 + 3 = 2
Therefore, the vertex is located at (-1, 2).
2. **Axis of Symmetry:**
The axis of symmetry for this function is the line x = -1, which runs vertically through the vertex.
3. **Y-Intercept:**
To find the y-intercept, we set x to 0:
f(0) = 0² + 2(0) + 3 = 3
This means the graph crosses the y-axis at (0, 3).
4. **Direction of the Parabola:**
Since the coefficient of x² (which is 1) is positive, we know that the parabola opens upwards.
Putting all of this information together, we can sketch the graph. The vertex is at (-1, 2), the parabola opens upwards, and it crosses the y-axis at (0, 3). The graph will have a U-shape with the lowest point at the vertex.
This information allows us to visualize what the graph of the function f(x) = x² + 2x + 3 looks like.