To graph the function f(x) = x² + 4x + 5, follow these steps:
1. Understand the Function:
This is a quadratic function in the form of f(x) = ax² + bx + c, where:
- a = 1
- b = 4
- c = 5
Since a > 0, the parabola opens upwards.
2. Find the Vertex:
The vertex of a quadratic function can be found using the formula:
x = -b / (2a)
Substituting the values of a and b:
x = -4 / (2 * 1) = -2
Now, substitute x = -2 back into the original function to find the y-coordinate:
f(-2) = (-2)² + 4(-2) + 5 = 4 – 8 + 5 = 1
So, the vertex is at the point (-2, 1).
3. Find the Y-Intercept:
The y-intercept is calculated by setting x = 0:
f(0) = 0² + 4(0) + 5 = 5
This means the y-intercept is at the point (0, 5).
4. Find the X-Intercepts:
To find the x-intercepts, set f(x) = 0:
x² + 4x + 5 = 0
This quadratic does not factor easily, so we can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)
Substituting the values:
b² – 4ac = 4² – 4(1)(5) = 16 – 20 = -4
Since the discriminant is negative (-4), it means there are no real x-intercepts; instead, the parabola lies completely above the x-axis.
5. Plotting Points:
To graph the function accurately, you can plot additional points:
- For x = -3: f(-3) = 9 – 12 + 5 = 2 → Point: (-3, 2)
- For x = -1: f(-1) = 1 – 4 + 5 = 2 → Point: (-1, 2)
- For x = 1: f(1) = 1 + 4 + 5 = 10 → Point: (1, 10)
6. Sketch the Graph:
Now that you have:
- Vertex: (-2, 1)
- Y-Intercept: (0, 5)
- Additional Points: (-3, 2), (-1, 2), (1, 10)
You can draw the parabola. Start by marking the vertex, y-intercept, and the points you plotted. Then, sketch a smooth curve through these points, ensuring the parabola opens upwards.
Conclusion:
By identifying the key features of the function f(x) = x² + 4x + 5, you can create an accurate graph that reflects its behavior. Enjoy the process of visualizing this quadratic function!