Functions can be used to model real-world events and interpret data from those events. Data that measures or compares two characteristics of a population is known as **bivariate data**.

When analyzing data, we previously described the relationship between two variables as linear or nonlinear. In this lesson, we will focus on nonlinear relationships that can be modeled by a quadratic function.

Each table shown represents a different set of data.

x | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 | 1.2 | 1.4 | 1.6 | 1.8 |
---|---|---|---|---|---|---|---|---|---|---|

y | 13 | 7 | 4 | 3 | 1 | 0 | 2 | 3 | 6 | 9 |

x | 3 | 3.5 | 4 | 4.5 | 5 | 5.5 | 6 | 6.5 | 7 | 7.5 | 8 |
---|---|---|---|---|---|---|---|---|---|---|---|

y | 63 | 68 | 77 | 90 | 104 | 100 | 112 | 120 | 114 | 127 | 127 |

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

y | 63 | 65 | 61 | 59 | 58 | 59 | 54 | 55 | 53 | 52 | 50 |

x | 1 | 2 | 2.5 | 3 | 4 | 5 | 6.3 | 6.8 | 7.2 | 7.4 | 8 |
---|---|---|---|---|---|---|---|---|---|---|---|

y | 1.5 | 3 | 4.8 | 5 | 7.4 | 8 | 7 | 6 | 5.5 | 4 | 2 |

Without creating a scatterplot:

Does the data in Table 1 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 2 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 3 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 4 have a linear or quadratic relationship? Explain your answer.

If points are more tightly clustered along the model, it represents a stronger relationship between the variables.

The curve of best fit can help us make predictions or conclusions about the data. If we are given an x-value, we can predict the y-value by substituting x into the equation and solving for y.

We can also use the graph of the model to approximate x and y-values.

When x=8,\,y\approx 3

When y=7,\,x\approx 4 and 6

When anayzing the data, it is often helpful to interpret the x-intercepts or the vertex in context. For example, if the equation models a company's sales over time, the x-intercepts represent the times the company made no sales, and the vertex represents the time the highest amount of sales were made.

It is important to consider the context of the data when communicating results as the model may only be appropriate over a part of the domain.

For each scatterplot, determine whether the variables have a linear relationship or a quadratic relationship. If there is a relationship, describe its strength.

a

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b

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c

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A conservationist tracks the population, y, of manatees that regularly visit a river over a number of years, x, (starting at zero). The data is displayed in the table:

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

y | 65 | 61 | 58 | 60 | 66 | 74 | 90 |

a

Was the data most likely collected through measurement, observation, a survey or an experiment?

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b

Determine if the manatee population over time has a quadratic relationship.

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c

Using technology, determine an appropriate equation to model the data set. Round all values to two decimal places.

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d

Using the model in part (b), determine the population 10 years afer the numbers were first recorded.

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Carlos is a goalie on the school soccer team. When he kicks a soccer ball dropped from his hands, he notices that the angle of trajectory for each kick is different. He also notices that there are times when the ball does not travel as far as other times. He wants to investigate this further using the data cycle.

a

Formulate a statistical question that Carlos can use for his investigation.

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b

Determine what variables could be used to answer the statistical question formulated in part (a).

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c

Carlos records 10 kicks and analyzes them to determine the angle of trajectory and also the distance traveled. His results are recorded in the table:

Angle (degrees) | 24 | 30 | 33 | 37 | 43 | 48 | 51 | 56 | 60 | 64 |
---|---|---|---|---|---|---|---|---|---|---|

Distance (feet) | 112 | 129 | 138 | 155 | 161 | 164 | 158 | 148 | 134 | 124 |

Determine if the data suggests a linear or quadratic relationship. Explain your answer.

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d

Using technology, determine an appropriate equation to model the data set.

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e

Draw a conclusion about the data by answering the statistical question from part (a).

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Idea summary

Data presents a *quadratic relationship* if it forms a symmetric curve or parabolic shape.

If points are more tightly clustered along the model, it represents a stronger relationship between the variables.