Lorraine writes the equation shown.
[tex]$x^2+y-15=0$[/tex]
She wants to describe the equation using the term relation and the term function.
The equation represents $\square$
Answer (1)
Rewrite the equation as y = − x 2 + 15 .
Verify that for each x value, there is only one corresponding y value.
Conclude that the equation represents a function.
State that since every function is also a relation, the equation represents both a relation and a function. Therefore, the equation represents a relation and a function. a re l a t i o nan d a f u n c t i o n
Explanation
Understanding the Problem The given equation is x 2 + y − 15 = 0 . We need to determine whether this equation represents a relation and/or a function.
Isolating y To determine if the equation represents a function, we need to check if for every value of x , there is only one corresponding value of y . Let's rewrite the equation to isolate y in terms of x .
Rewriting the Equation Adding 15 − x 2 to both sides of the equation, we get:
y = − x 2 + 15
Checking for Function Now, we can see that for any value of x , there is only one corresponding value of y . For example, if x = 0 , then y = − 0 2 + 15 = 15 . If x = 1 , then y = − 1 2 + 15 = 14 . If x = − 1 , then y = − ( − 1 ) 2 + 15 = 14 . Since each value of x gives a unique value of y , the equation represents a function.
Conclusion Since every function is also a relation, the equation also represents a relation. Therefore, the equation represents both a relation and a function.
Examples
In physics, the equation y = − x 2 + 15 could describe the height y of a projectile at a horizontal distance x from the launch point, assuming a parabolic trajectory. Understanding that this equation represents a function allows us to predict the height of the projectile at any given horizontal distance. This concept is crucial in fields like sports science, where analyzing trajectories helps athletes optimize their performance, and in engineering, where designing efficient projectile systems is essential.
