To find the x and y intercepts of a quadratic function, we follow a straightforward process. A quadratic function is typically in the form of f(x) = ax² + bx + c. Here’s how to find both intercepts:
Finding the Y-Intercept
The y-intercept is found by determining the value of the function when x = 0. This means substituting 0 in place of x in the function:
f(0) = a(0)² + b(0) + c = c
So, the y-intercept is the point (0, c). For example, if your quadratic function is f(x) = 2x² + 3x + 5, the y-intercept would be (0, 5).
Finding the X-Intercepts
X-intercepts occur where the graph of the function crosses the x-axis. This happens when f(x) = 0. To find the x-intercepts, set the quadratic equation equal to zero:
0 = ax² + bx + c
You can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
The values of x you get from this formula will give you the x-intercepts, which are the points (x₁, 0) and (x₂, 0). They can be real or complex numbers depending on the value of the discriminant, (b² – 4ac).
Example
Consider the quadratic function:
f(x) = x² – 5x + 6
First, to find the y-intercept, we substitute x = 0:
f(0) = 0² – 5(0) + 6 = 6, so the y-intercept is (0, 6).
Next, to find the x-intercepts:
0 = x² – 5x + 6
Using the quadratic formula:
x = (5 ± √((-5)² – 4(1)(6))) / 2(1)
Calculating further, we find:
x = (5 ± √(1)) / 2
This results in x = 3 and x = 2. Hence, the x-intercepts are (2, 0) and (3, 0).
In summary, the y-intercept is found by substituting zero for x whereas x-intercepts are found by solving the equation for when f(x) equals zero. This approach gives you a complete picture of where the quadratic function meets the axes on the graph.