Question 22·SAT Desmos Regression Practice·Advanced Math

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The quadratic function $h$ models the height, in meters, of a firework above the ground $t$ seconds after it is launched.

The model estimates that the firework reached its maximum height of $200$ meters $5$ seconds after it was launched. At the moment of launch, the firework was $75$ meters above the ground.

Based on the model, what was the estimated height, to the nearest meter, of the firework $7$ seconds after it was launched?

Your answer

A described maximum is the vertex, the highest point of a quadratic graph. It gives the time and height to use in vertex form. Use the starting height to find the remaining coefficient with a one-unknown Desmos regression, then evaluate at the requested time. The height below the maximum grows with the square of the time from the peak, not at a constant rate.

Hints

Hint 1
A quadratic with its vertex at time $5$ and height $200$ has the form $h(t)=a(t-5)^2+200$ . The squared part is zero at $t=5$ . What given height can you use to find $a$ ?
Hint 2
At launch, $t=0$ , so the starting height means $h(0)=75$ . Put those numbers into vertex form. The launch point gives new information; putting in the vertex instead would leave $a$ unknown.
Hint 3
Once you've found $a$ , evaluate $h(7)$ . The input $7$ means seconds after launch, while the output is the height in meters.

Step-by-step

Approach 1: Build the model in vertex form
Step 1
Use the maximum to write vertex form
The vertex is a quadratic graph's turning point. Here the maximum occurs at $t=5$ with height $200$ , so use a squared term that becomes zero at $t=5$ :
$h(t)=a(t-5)^2+200.$
The number $a$ is still unknown; the maximum alone can't tell you how quickly the height falls.
Step 2
Turn the launch height into an equation
At launch, $t=0$ and the height is $75$ meters, so $h(0)=75$ . Put those values into the model:
$75=a(0-5)^2+200.$
Unlike the vertex, this point can determine $a$ because its squared term isn't zero.
Step 3
Find the missing coefficient
Type $75\sim a(0-5)^2+200$ . The tilde asks Desmos to fit the unknown $a$ to the launch height. Under PARAMETERS, it shows $a=-5$ . The negative sign makes the height fall on either side of the maximum.
Step 4
Define the height function
Type $h(t)=a(t-5)^2+200$ . Desmos uses the stored value $a=-5$ , so this defines $h(t)=-5(t-5)^2+200$ . Now you can give the function the requested time.
Step 5
Find the height after seven seconds
Type $h(7)$ , which means put $7$ in for the time. Desmos shows $180$ . That's the firework's estimated height, not the distance it has fallen from the maximum. To the nearest meter, the height is $180$ meters. Grid in 180.

Approach 2: Scale the drop from the maximum
Step 1
Find the drop at launch
The firework starts at $75$ meters and reaches a maximum of $200$ meters, so at launch it is $125$ meters below the peak:
$200-75=125.$
Step 2
Compare distances from the peak
Launch at $t=0$ is $5$ seconds from the peak at $t=5$ ; the requested time, $t=7$ , is $2$ seconds from it. In vertex form, the drop from the peak scales with the squared distance in time, so the new drop is $125\left(\frac{2}{5}\right)^2$ . Use $2$ seconds, not $7$ , as the distance from the peak.
Step 3
Subtract the drop from the maximum
Type $200-125(2/5)^2$ . Desmos shows $180$ . This subtracts the drop below the peak from the maximum, giving an estimated height of $180$ meters after $7$ seconds. Grid in 180.

Question 22·SAT Desmos Regression Practice·Advanced Math

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