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Perplex
Perplex
Dashboard
Topics
Exponents & LogarithmsApproximations & ErrorSequences & SeriesFinancial Mathematics
Cartesian plane & linesFunction TheoryModelling
2D & 3D GeometryVoronoi Diagrams
ProbabilityDescriptive StatisticsBivariate StatisticsDistributions & Random VariablesInference & Hypotheses
DifferentiationIntegration
Review VideosFormula BookletMy Progress
BlogLanding Page
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Perplex

Modelling (Lesson 3/7)

Quadratic Models

1 / 11

Discussion

Jessica, a third student in Dr. Tyler's class, wants to model the relationship between the height of a tennis ball she throws into the air as a function of time. She plots her data on the graph below:

Describe a curve that could model the motion of the tennis ball. Note any graphical features of the curve that distinguish the model from a straight line.

Solution:

A suitable model for the motion of the tennis ball is a smooth, symmetric curve shaped like an "upside-down U" (a parabola). This curve starts at the point ​(0,0), rises to a single maximum point near ​(1,5), and then descends back to ​(2,0), matching the pattern of the plotted data.

This curve is different from a straight line in several important ways:

  • The slope (rate of change) is positive for ​t<1, meaning the height increases as time increases.

  • At the top of the curve, around ​t≈1, the slope is zero. This is called a "turning point" or "vertex," where the ball reaches its maximum height before starting to fall.

  • For ​t>1, the slope is negative, so the height decreases as time increases.

A straight line would have a constant slope and could not model the ball's rise and fall. The presence of a single turning point and the change in slope are key features that distinguish this curve from a straight line.

Modelling (Lesson 3/7)

Quadratic Models

1 / 11

Discussion

Jessica, a third student in Dr. Tyler's class, wants to model the relationship between the height of a tennis ball she throws into the air as a function of time. She plots her data on the graph below:

Describe a curve that could model the motion of the tennis ball. Note any graphical features of the curve that distinguish the model from a straight line.

Solution:

A suitable model for the motion of the tennis ball is a smooth, symmetric curve shaped like an "upside-down U" (a parabola). This curve starts at the point ​(0,0), rises to a single maximum point near ​(1,5), and then descends back to ​(2,0), matching the pattern of the plotted data.

This curve is different from a straight line in several important ways:

  • The slope (rate of change) is positive for ​t<1, meaning the height increases as time increases.

  • At the top of the curve, around ​t≈1, the slope is zero. This is called a "turning point" or "vertex," where the ball reaches its maximum height before starting to fall.

  • For ​t>1, the slope is negative, so the height decreases as time increases.

A straight line would have a constant slope and could not model the ball's rise and fall. The presence of a single turning point and the change in slope are key features that distinguish this curve from a straight line.