Understanding Parabolas: How to Identify Zeros of a Function

In mathematics, understanding the concept of zeros of a function is crucial, especially when dealing with parabolas. A parabola is a U-shaped curve that can open either upwards or downwards, and its zeros are the points where the graph intersects the x-axis. In our case, the given parabola crosses the x-axis at two points: x = 3 and x = 7. These points are called the 'zeros' or 'roots' of the function, meaning that when you substitute these x-values into the equation, the result will be zero.

To illustrate this with the equation provided, we have:

**y = -(x - 3)(x - 7)**.

This equation is in factored form, which is particularly useful for finding zeros. When you set y = 0, you can solve for x:

**0 = -(x - 3)(x - 7)**.

This equation will be true if either (x - 3) = 0 or (x - 7) = 0. Solving these gives us x = 3 and x = 7, confirming that these points are indeed the zeros of the function.

The negative sign in front of the equation indicates that the parabola opens downward. This means that the highest point, or vertex, of the parabola will be above the x-axis, and the curve will slope downwards towards the x-axis at x = 3 and x = 7. A downward-opening parabola typically represents a maximum point, which can be important in various applications, such as in physics for projectile motion.

For example, if you're modeling the path of a ball thrown into the air, the highest point the ball reaches corresponds to the vertex of the parabola. Understanding how to find zeros helps in predicting the motion of the ball as it goes up and comes back down, hitting the ground at specific points (the zeros).

To further reinforce this concept, let's consider another example:

If we have the equation **y = -(x + 2)(x - 4)**, we can find the zeros by setting y = 0:

**0 = -(x + 2)(x - 4)**.

Here, solving gives us x = -2 and x = 4 as the zeros. Again, the negative sign indicates the parabola opens downward, and understanding these properties can be applied in various real-world scenarios, from engineering to economics.

In summary, recognizing the zeros of a parabola not only aids in graphing the function accurately but also enhances your ability to model and solve real-life problems effectively. Always remember to check the equations section for additional insights on how zeros are derived from the factored form of equations.