Question 887·Hard·Math

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The scatterplot shows the height $y$ (in feet) of a soccer ball $x$ seconds after it was kicked.

Which choice gives a quadratic function that best models the data in the scatterplot?

A quadratic model follows a parabola, so look for heights that match at times on opposite sides of its peak. Put the plotted points in a Desmos table and fit a quadratic in vertex form, which shows the peak and the curve's bend. Don't treat the highest measured dots as the exact peak; it may lie between them.

Hints

Hint 1
A parabola is symmetric: points equally far from its middle have the same height. Pair the plotted times that share a height. Where is the middle of each pair?
Hint 2
The axis of symmetry is the vertical line through a parabola's peak. The two highest plotted points are at neighboring times, so their midpoint gives the time of the peak, not necessarily its height.
Hint 3
Fit vertex form, $y=a(x-h)^2+k$ , to the points in Desmos. The fitted $h$ gives the peak's time, $k$ gives its height, and $a$ controls the bend.

Step-by-step

Fit the plotted points in Desmos
Step 1
Enter the points and spot the matching heights
Enter the six plotted points in a Desmos table, with time in $x_1$ and height in $y_1$ . Each row keeps one time with its height. The table shows equal-height pairs at times $0$ and $5$ , $1$ and $4$ , and $2$ and $3$ . Those pairs point to a symmetric parabola.
Step 2
Find the time halfway between the highest dots
The highest plotted dots occur at $x=2$ and $x=3$ . A parabola mirrors across its axis of symmetry, so that axis lies halfway between them. Type their average, $\frac{2+3}{2}$ , in Desmos. It shows $2.5$ , or $\frac52$ seconds. That is the peak's time, not its height.
Step 3
Fit the parabola and choose its equation
Type $y_1\sim a(x_1-h)^2+k$ . The $\sim$ tells Desmos to find the best-fit values; in this form, $h$ is the peak's time, $k$ its height, and $a$ sets the bend. Under PARAMETERS, Desmos shows $a=-2$ , $h=2.5$ , and $k=15$ . So the model is $y=-2\left(x-\frac52\right)^2+15$ . It predicts a peak height of $15$ feet at $2.5$ seconds. The highest measured dots need not be the model's peak. Choice A.

Question 887·Hard·Math

Strategy

Hints

Step-by-step

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