Graphing the function f(x) = x² + 4x + 5 involves a few steps to understand its shape and position.
1. Identify the type of function: This is a quadratic function, which means its graph will be a parabola. Since the coefficient of x² is positive, the parabola opens upwards.
2. Find the vertex: To find the vertex of the parabola, use the vertex formula, which is given by x = -b/(2a). Here, a = 1 and b = 4.
Calculating this gives:
x = -4/(2 * 1) = -2
Now, substitute x = -2 back into the function to find the y-coordinate of the vertex:
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: To find the y-intercept, set x = 0:
f(0) = 0² + 4(0) + 5 = 5
Thus, 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
Using the determinant method, b² – 4ac:
b² – 4ac = 4² – 4(1)(5) = 16 – 20 = -4
Since the determinant is negative, there are no real x-intercepts, meaning the graph does not cross the x-axis.
5. Sketch the graph: Now that you have the vertex (-2, 1), the y-intercept (0, 5), and knowing that the parabola opens upwards with no x-intercept, you can sketch the graph:
Start at the vertex, plot the y-intercept, and draw a smooth curve that opens upward, ensuring it does not cross the x-axis.
This simple process helps visualize the graph of the quadratic function f(x) = x² + 4x + 5.