The table below gives the relation between $x$ and $y$ for a parabola.
| X | 4 | 5 | 6 | 7 |
|---|---|---|---|---|
| Y | 32 | 50 | 72 | 98 |
Which equation describes the relation?
Answer (2)
Analyze the given data points ( 4 , 32 ) , ( 5 , 50 ) , ( 6 , 72 ) , and ( 7 , 98 ) .
Test the equation y = 2 x 2 for each point.
Verify that the equation holds true for all given points.
Conclude that the equation describing the relation is y = 2 x 2 .
Explanation
Understanding the Problem We are given a table of x and y values that represent a parabola, and we need to find the equation that describes this relationship.
Analyzing the Data Let's examine the given data points: ( 4 , 32 ) , ( 5 , 50 ) , ( 6 , 72 ) , and ( 7 , 98 ) . We will look for a pattern to determine the equation.
Testing a Potential Equation Notice that when x = 4 , y = 32 . If we try y = 2 x 2 , we get y = 2 ( 4 2 ) = 2 ( 16 ) = 32 . This holds true for the first point. Let's check the other points as well.
Verifying the Equation When x = 5 , y = 50 . Using y = 2 x 2 , we get y = 2 ( 5 2 ) = 2 ( 25 ) = 50 . This also holds true. When x = 6 , y = 72 . Using y = 2 x 2 , we get y = 2 ( 6 2 ) = 2 ( 36 ) = 72 . This is also correct. When x = 7 , y = 98 . Using y = 2 x 2 , we get y = 2 ( 7 2 ) = 2 ( 49 ) = 98 . This is also correct.
Conclusion Since the equation y = 2 x 2 holds true for all the given points, we can conclude that this is the equation that describes the relation between x and y .
Examples
Understanding parabolic relationships is crucial in various real-world applications. For instance, the trajectory of a projectile, like a ball thrown in the air, follows a parabolic path. The equation y = 2 x 2 can be used to model simplified scenarios where the height y of the ball depends on the horizontal distance x from the thrower, ignoring air resistance and other factors. Similarly, the shape of satellite dishes and reflecting telescopes are based on parabolic curves, which help focus signals or light to a single point, enhancing their efficiency.
The equation describing the relationship between x and y in the given table is y = 2 x 2 . This was verified by substituting each value of x into the equation, where it accurately produced the corresponding values of y . Therefore, the function describes a parabola based on the data points provided.
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