The real zeros of a parabola are the x-coordinates where the graph of the parabola intersects or touches the horizontal x-axis. These points represent the solutions to the quadratic equation that defines the parabola when the function's output (y-value) is zero.
The graph of a quadratic function is known as a parabola. The "zeros" of a parabola, also frequently referred to as roots or x-intercepts, are specifically the points on the parabola that lie on the line y = 0, which is the x-axis itself. When we speak of real zeros, we are referring to those solutions that are real numbers, meaning they can be plotted on a standard coordinate plane.
Understanding Parabola Zeros
A parabola is the distinctive U-shaped or inverted U-shaped curve formed by the graph of a quadratic function, typically written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a is not equal to zero.
The real zeros indicate where the parabola crosses or touches the x-axis. These are crucial points for understanding the behavior of the quadratic function.
How Many Real Zeros Can a Parabola Have?
A parabola can have a maximum of two real zeros. The number of real zeros depends on its position relative to the x-axis. There are three possible scenarios:
- Two Distinct Real Zeros: This occurs when the parabola crosses the x-axis at two separate points. The quadratic equation has two unique real solutions.
- One Real Zero (Repeated Root): The parabola touches the x-axis at exactly one point, which is its vertex. This means the quadratic equation has one real solution that is repeated.
- No Real Zeros: The parabola does not intersect the x-axis at all. It might be entirely above the x-axis (opening upwards) or entirely below it (opening downwards). In this case, the solutions to the quadratic equation are complex (or imaginary) numbers, not real numbers.
Finding Real Zeros
There are several methods to find the real zeros of a parabola:
- Factoring: If the quadratic expression can be factored, set each factor equal to zero and solve for x.
- Quadratic Formula: This is a universal method that works for any quadratic equation. For ax² + bx + c = 0, the solutions for x are given by:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ - Graphing: By plotting the parabola, you can visually identify the points where it intersects the x-axis.
The Discriminant: Determining the Number of Real Zeros
The discriminant, which is the expression b² - 4ac found under the square root in the quadratic formula, provides a quick way to determine the nature and number of the zeros without fully solving the equation.
| Discriminant Value (D) | Number of Real Zeros | Parabola's Interaction with X-axis |
|---|---|---|
| D > 0 | Two distinct real zeros | Crosses at two separate points |
| D = 0 | One real zero (repeated) | Touches the x-axis at its vertex |
| D < 0 | No real zeros | Does not intersect the x-axis |
Understanding the discriminant is key to predicting how many real solutions a quadratic equation has, and thus, how many times its corresponding parabola will cross or touch the x-axis. For more details on the quadratic formula and discriminant, you can refer to resources like Khan Academy's explanation of the quadratic formula.
Practical Example
Let's find the real zeros of the parabola defined by the equation f(x) = x² - 5x + 6.
Here, a = 1, b = -5, and c = 6.
-
Calculate the Discriminant:
D = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1.
Since D = 1 (which is > 0), we know there will be two distinct real zeros. -
Use the Quadratic Formula:
$x = \frac{-(-5) \pm \sqrt{1}}{2(1)}$
$x = \frac{5 \pm 1}{2}$This gives us two solutions:
- $x_1 = \frac{5 + 1}{2} = \frac{6}{2} = 3$
- $x_2 = \frac{5 - 1}{2} = \frac{4}{2} = 2$
Therefore, the real zeros of the parabola f(x) = x² - 5x + 6 are x = 2 and x = 3. This means the parabola intersects the x-axis at the points (2, 0) and (3, 0).
Why Are Real Zeros Important?
Real zeros are fundamental in many real-world applications of quadratic functions:
- Projectile Motion: In physics, the zeros of a parabolic trajectory indicate when a thrown object (like a ball) hits the ground (height = 0).
- Optimization Problems: In business or engineering, finding zeros can help identify break-even points, maximum profits, or minimum costs.
- Design and Engineering: Understanding where structures or paths intersect a baseline can be crucial for design.
In summary, the real zeros of a parabola provide critical information about where the function's value is zero, corresponding to the points where its graph crosses or touches the x-axis.
