The zeros of a quadratic function are the $x$-values where the graph of the quadratic function crosses the $x$-axis. Move the sliders for $a$ and $b$, and notice how the $x$-intercepts, or zeros, always match the values of $a$ and $b$.
$y=(x-a)(x-b)$
a. 0
b. 0
Answer (2)
Set the quadratic function equal to zero: ( x − a ) ( x − b ) = 0 .
Apply the zero product property: either x − a = 0 or x − b = 0 .
Solve for x in each case: x = a or x = b .
The zeros of the quadratic function are x = a and x = b : x = a , x = b .
Explanation
Understanding the Zeros of a Quadratic Function We are given a quadratic function in the form y = ( x − a ) ( x − b ) and we want to explain why the zeros of this function are x = a and x = b . The zeros of a function are the x -values where the graph of the function intersects the x -axis, which means y = 0 at those points.
Setting y=0 To find the zeros, we set y = 0 in the equation y = ( x − a ) ( x − b ) . This gives us the equation ( x − a ) ( x − b ) = 0 .
Applying the Zero Product Property For the product of two factors to be equal to zero, at least one of the factors must be equal to zero. Therefore, either ( x − a ) = 0 or ( x − b ) = 0 .
Solving for x If x − a = 0 , then by adding a to both sides of the equation, we get x = a . Similarly, if x − b = 0 , then by adding b to both sides of the equation, we get x = b .
Conclusion Thus, the zeros of the quadratic function y = ( x − a ) ( x − b ) are x = a and x = b . This explains why, when you move the sliders for a and b , the x -intercepts of the graph always match the values of a and b .
Examples
Understanding the zeros of a quadratic function is crucial in many real-world applications. For instance, if you are designing a bridge with a parabolic arch, the zeros of the quadratic function describing the arch's shape would tell you where the arch meets the ground. Similarly, in physics, if you are analyzing the trajectory of a projectile, the zeros of the quadratic function describing its height would tell you when the projectile hits the ground. Knowing the zeros allows engineers and scientists to predict outcomes and design structures and systems effectively.
The zeros of the quadratic function y = ( x − a ) ( x − b ) occur at x = a and x = b , found by solving the equation ( x − a ) ( x − b ) = 0 . This demonstrates where the graph intersects the x-axis. When adjusting the values for a and b , the x-intercepts shift accordingly, highlighting their significance.
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