Question 57·Hard·Two-Variable Data: Models and Scatterplots
The scatterplot shows the height (in feet) of a soccer ball seconds after it was kicked.
Which choice gives a quadratic function that best models the data in the scatterplot?
For a scatterplot that rises to a maximum and then falls, use vertex form . Read and from the location of the peak (often by using symmetry), then plug in one additional point to solve for . Finally, verify the model against another point to confirm the best match.
Hints
Look at the overall shape
If the data rise to a peak and then fall, a quadratic model (a parabola) is often a good fit.
Estimate the vertex from the highest point
Find the approximate maximum -value and the -value where it occurs; that gives the vertex for vertex form.
Use vertex form to solve for the vertical stretch
After writing , plug in one other plotted point to solve for .
Desmos Guide
Enter the data points
In Desmos, enter the plotted points: (0,2.5), (1,10.5), (2,14.5), (3,14.5), (4,10.5), (5,2.5).
Graph each answer choice
Enter each quadratic from the choices and display all graphs on the same axes.
Decide which graph best matches the scatterplot
Check which graph has its peak near and stays closest to the points across to .
Step-by-step Explanation
Identify the model type and the vertex
The points increase and then decrease, forming a curved shape consistent with a downward-opening parabola.
The maximum height is about feet and occurs halfway between and , so the vertex is approximately
Write the quadratic in vertex form
Use vertex form
With and :
Use a plotted point to find
Use the point near :
So
Select the matching equation
Substitute into the vertex form:
So the correct choice is .