Louis used a quadratic equation to model the height, $y$, of a falling object $x$ seconds after it is dropped. Which ordered pair generated by this model should be discarded because the values are unreasonable?
A. $(-4,1)$
B. $\left(\frac{9}{4}, 24\right)$
C. $\left(5, \frac{3}{2}\right)$
D. (6, 3)
Answer (2)
The problem provides four ordered pairs representing the height of a falling object at different times.
Time cannot be negative, so any ordered pair with a negative x-value is unreasonable.
The ordered pair ( − 4 , 1 ) has a negative x-value, making it unreasonable.
Therefore, the ordered pair that should be discarded is ( − 4 , 1 ) .
Explanation
Understanding the Problem We are given four ordered pairs that represent the height, y , of a falling object x seconds after it is dropped. We need to determine which ordered pair is unreasonable.
Checking the First Ordered Pair The first ordered pair is ( − 4 , 1 ) . Since x represents time, it cannot be negative. Therefore, this ordered pair is unreasonable.
Checking the Other Ordered Pairs The other ordered pairs are ( 4 9 , 24 ) , ( 5 , 2 3 ) , and ( 6 , 3 ) . The x values are all positive, so they are reasonable in terms of time.
Considering the Context of a Falling Object As time increases, the height of a falling object should decrease. Let's compare the last two ordered pairs: ( 5 , 2 3 ) and ( 6 , 3 ) . As x increases from 5 to 6, y increases from 2 3 to 3. This is unreasonable because the height should decrease as time increases. However, the ordered pair ( − 4 , 1 ) is still more obviously unreasonable because time cannot be negative.
Final Answer Therefore, the ordered pair that should be discarded is ( − 4 , 1 ) .
Examples
Understanding quadratic equations helps us model various real-world scenarios, such as the trajectory of a ball thrown in the air or the height of a rocket launched into space. In these situations, the x-value represents time, and the y-value represents the height of the object. Recognizing that time cannot be negative is crucial for interpreting the model correctly and discarding unreasonable data points. This ensures that our predictions are physically meaningful and accurate.
The ordered pair ( − 4 , 1 ) should be discarded because it contains a negative time value, which is unreasonable in the context of a falling object. All other pairs involve positive time values. Thus, ( − 4 , 1 ) is the only option that does not fit the physical model of a falling object.
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