Question 127·Easy·Nonlinear Functions
A firework is launched from a platform 1 meter above the ground. The height, in meters, of the firework seconds after launch is modeled by
Which of the following is the best interpretation of in this context?
For questions that ask you to interpret something like in context, first identify what the input and output of the function represent (including units). Then read as “when the input is [with its units], the output is [with its units].” Finally, eliminate choices that (1) swap the numbers, (2) turn the value into a rate (like “per second”) when it’s not, or (3) claim something stronger, like a maximum or minimum, that isn’t directly stated by that function value.
Hints
Think about inputs and outputs
Ask yourself: in the function , which quantity is the input, and which quantity is the output? What do they represent in the story?
Focus on
means you plug in . What does measure here, and what does measure? Keep track of their units (seconds and meters).
Check the order of the numbers
The pair is associated with . Which answer choice talks about 3 as the time and 16 as the height, and not the other way around? Also, be careful not to confuse a single height value with a rate or with a maximum.
Desmos Guide
Graph the height function
In Desmos, type the equation y = -5x^2 + 20x + 1 to graph the firework’s height as a function of time (with as time in seconds and as height in meters).
Find the value at 3 seconds
Add a table in Desmos, put 3 in the -column, and look at the corresponding -value in the table (this is ). You can also click on the graph at and read the coordinate from the point label.
Interpret the coordinate in context
Take the coordinate you found for and interpret it as (time in seconds, height in meters). Choose the answer option that states this time and height in words, in the correct order and with the correct units.
Step-by-step Explanation
Understand what the function represents
The function
gives the height of the firework in meters. The input is time in seconds after launch, and the output is height above the ground in meters at that time.
Interpret what means mathematically
The expression means to plug in into the function.
Compute it to see the value:
So is the ordered pair on the graph of the function.
Match numbers to their real-world meanings
In this context:
- The input has units of seconds (time after launch).
- The output has units of meters (height above the ground).
So the correct interpretation must say something like: time = 3 seconds and height = 16 meters, in that order (time first, then height), and must not talk about a rate (like 'per second') or a maximum unless that is clearly stated by another part of the function.
Choose the answer that matches the interpretation
Check each option:
- A) Switches the roles of 3 and 16 (it uses 16 seconds and 3 meters), so it does not match .
- B) Describes a rate (3 meters every 16 seconds), but is a single height at one time, not a speed.
- C) Claims 16 meters is the maximum height, but is just one height value; nothing here says it is the highest point.
- D) Correctly states that the time is 3 seconds and the height is 16 meters.
Therefore, the best interpretation is: At 3 seconds after launch, the firework is 16 meters above the ground.