The graph of the function f(x) = x² + 2x + 3 is a parabola that opens upwards. To better understand its shape and key features, let’s break it down:
1. Standard Form
The function is in the standard quadratic form, which is represented as f(x) = ax² + bx + c. In this case:
- a = 1
- b = 2
- c = 3
This means that the parabola opens upward since a (1) is positive.
2. Vertex of the Parabola
The vertex of a parabola given by the equation y = ax² + bx + c can be found using the formula:
x = -b/(2a). Plugging in our values:
- x = -2/(2 * 1) = -1
Next, we can find the corresponding y value by substituting x = -1 back into the function:
- f(-1) = (-1)² + 2(-1) + 3 = 1 – 2 + 3 = 2
Thus, the vertex of the parabola is at the point (-1, 2).
3. Y-Intercept
The y-intercept occurs where x = 0:
- f(0) = 0² + 2(0) + 3 = 3
This means the y-intercept is at the point (0, 3).
4. Symmetry
The parabola is symmetric about the vertical line that passes through the vertex. In this case, the line of symmetry is x = -1.
5. Graphing
To graph the function accurately:
- Plot the vertex at (-1, 2).
- Plot the y-intercept at (0, 3).
- Choose additional x-values to find corresponding y-values:
- f(-2) = (-2)² + 2(-2) + 3 = 1 at point (-2, 1)
- f(1) = (1)² + 2(1) + 3 = 6 at point (1, 6)
Connecting these points will form a smooth curve, showing the characteristic U-shape of the parabola.
Conclusion
In summary, the graph of the function f(x) = x² + 2x + 3 is a U-shaped parabola that opens upwards with its vertex located at (-1, 2) and a y-intercept at (0, 3). By plotting additional points and connecting them smoothly, you will be able to visualize this quadratic function effectively!