What are the zeros of the graphed function $y=\frac{1}{3}(x+3)(x+5)$?
-5 and -3
5 and 3
-8 and 5
0 and 5
Answer (1)
To find the zeros of the function:
Set the function equal to zero: 3 1 ( x + 3 ) ( x + 5 ) = 0 .
Recognize that the equation is satisfied when ( x + 3 ) ( x + 5 ) = 0 .
Solve for x by setting each factor to zero: x + 3 = 0 or x + 5 = 0 .
The zeros are − 5 and − 3 .
Explanation
Understanding the Problem We are given the function y = 3 1 ( x + 3 ) ( x + 5 ) and asked to find its zeros. The zeros of a function are the x -values for which y = 0 .
Setting y=0 To find the zeros, we set y = 0 and solve for x : 0 = 3 1 ( x + 3 ) ( x + 5 ) .
Simplifying the Equation Since 3 1 is a non-zero constant, the equation is satisfied when the product of the factors ( x + 3 ) and ( x + 5 ) is zero: ( x + 3 ) ( x + 5 ) = 0 .
Finding the Roots This equation is satisfied if either x + 3 = 0 or x + 5 = 0 . Let's solve each of these equations separately.
Solving for x (first root) If x + 3 = 0 , then subtracting 3 from both sides gives x = − 3 .
Solving for x (second root) If x + 5 = 0 , then subtracting 5 from both sides gives x = − 5 .
Final Answer Therefore, the zeros of the function are x = − 3 and x = − 5 .
Examples
Understanding zeros of a function is crucial in many real-world applications. For example, if the function represents the height of a projectile over time, the zeros would represent the times when the projectile hits the ground. Similarly, in business, if the function represents profit, the zeros would represent break-even points where the company neither makes nor loses money. Finding the zeros helps in analyzing and predicting the behavior of the system being modeled.
