Suppose you recorded the high temperature for each day of the year. If you wanted to model this data with a function, how would you decide whether to use a linear model, exponential model, or quadratic model. Could your graphing calculator help you decide? If so, what buttons would you have to push on your calculator in order to get relevant information?

### Linear, Exponential, and Quadratic Models

You should be familiar with how to graph three very important types of equations:

- Linear equations in slope-intercept form: \begin{align*}y=mx+b\end{align*}
- Exponential equations of the form: \begin{align*}y=a(b)^x\end{align*}
- Quadratic equations in standard form: \begin{align*}y=ax^2+bx+c\end{align*}

In real-world applications, the function that describes some physical situation is not given. Finding the function is an important part of solving problems. For example, scientific data such as observations of planetary motion are often collected as a set of measurements given in a table. One job for a scientist is to figure out which function best fits the data. Now, you will learn some methods that are used to identify which function describes the relationship between the dependent and independent variables in a problem.

#### Using Differences to Determine the Model

By finding the differences between the dependent values, we can determine the **degree** of the model for the data.

- If the first difference is the same value, the model will be
**linear.** - If the second difference is the same value, the model will be
**quadratic.** - If the number of times the difference has been taken before finding repeated values exceeds five, the model may be exponential or some other special equation.

#### Let's determine which model to use given the following tables of values:

The first difference (the difference between any two successive output values) is the same value (3). This means that this data can be modeled using a **linear regression line.**

The equation to represent this data is \begin{align*}y=3x+2\end{align*}.

This is a quadratic model because the second differences are the differences that have the same value (4). When we look at the difference of the \begin{align*}y-\end{align*}values, we must make sure that we examine entries for which the \begin{align*}x-\end{align*}values increase by the same amount.

#### Using Ratios to Determine the Model

Finding the differences involves subtracting the dependent values leading to the degree of the model. By taking the **ratio** of the values, one can determine whether the model is exponential.

If the ratio of dependent values is the same, then the data is modeled by an **exponential** equation.

#### Let's find the model that represents the following table of values:

Note that the ratio of values is the same between each set of numbers. This is an exponential equation.

The equation to represent this data is \begin{align*}y = 4 \cdot 3^x\end{align*}.

#### Determining the Model Using a Graphing Calculator

To enter data into your graphing calculator, find the **[STAT]** button. Choose **[EDIT].**

**[L1]**represents your independent variable, your \begin{align*}x\end{align*}.**[L2]**represents your dependent variable, your \begin{align*}y\end{align*}.

Enter the data into the appropriate list. Using the first set of data to illustrate yields:

You already know this data is best modeled by a linear regression line. Using the **[CALCULATE]** menu of your calculator, find the linear regression line, **LinReg.**

Look at the screen above. This is where you can find the **quadratic regression line [QUADREG]**, the **cubic regression line [CUBICREG]**, and the exponential regression line, **[EXPREG].**

### Examples

#### Example 1

Earlier, you were asked to determine which mathematical model was best to model the high temperatures for each day of this year. If you used a graphing calculator to help you decide, what buttons would you have to push?

You can use your graphing calculator to help you determine the best model for this data. In order to do this, follow the steps below.

- Press
**[STAT],**then**[EDIT].** - Next, enter your data into
**[****L1]**and**[L2].****[L1]**represents your independent variable, your \begin{align*}x\end{align*}.**[L2]**represents your dependent variable, your \begin{align*}y\end{align*}.

- Press
**[CALCULATE]**, then select**LinReg****[QUADREG],**the cubic regression line**[CUBICREG],**and the exponential regression line**[EXPREG].**Now you can look at each model and see which fits the best.

#### Example 2

Determine whether the function in the given table is linear, quadratic or exponential.

\begin{align*}& x & y \\ & 0 & 5\\ &1 & 10\\ &3 & 20\\ & 4 & 25\\ &6 & 35 \end{align*}

At first glance, this function might not look linear because the difference in the \begin{align*}y-\end{align*}values is not always the same.

However, we see that the difference in \begin{align*}y-\end{align*}values is 5 when we increase the \begin{align*}x-\end{align*}values by 1, and it is 10 when we increase the \begin{align*}x-\end{align*}values by 2. This means that the difference in \begin{align*}y-\end{align*}values is always 5 when we increase the \begin{align*}x-\end{align*}values by 1. Therefore, the function is linear.

The equation is modeled by \begin{align*}y=5x+5\end{align*}.

### Review

- The second set of differences have the same value. What can be concluded?
- Suppose you find the differences five different times and still don't come to a common value. What can you safely assume?
- Why would you test the ratio of differences?
- If you had a cubic (\begin{align*}3^{rd}\end{align*}-degree) function, what could you conclude about the differences?

Determine whether the data can be modeled by a linear equation, a quadratic equation, or neither.

- \begin{align*}& x && \text{-}4 && \text{-}3 && \text{-}2 && \text{-}1 && 0 && \qquad 1\\ & y && 10 && 7 && 4 && 1 && \! \text{-}2 && \qquad \! \text{-}5\end{align*}
- \begin{align*}\ \\ & x && \text{-}2 && \text{-}1 && 0 && 1 && 2 && \qquad 3\\ & y && 4 && 3 && 2 && 3 && 6 && \qquad \! 11\\ \ \\\end{align*}
- \begin{align*}& x && 0 && 1 && 2 && 3 && 4 && \qquad 5 \\ & y && \! 50 && \! 75 && \! \! \! 100 && \! \! \! 125 && \! \! \! 150 && \! \! \! \qquad 175\end{align*}
- \begin{align*}\ \\ & x && \text{-}10 && \text{-}5 && 0 && 5 && 10 && \qquad 15\\ & y && \, 10 && \! 2.5 && 0 && \! \! 2.5 && 10 && \qquad \! \! 22.5\\ \ \\\end{align*}
- \begin{align*}& x && 1 && 2 && 3 && 4 && 5 && \qquad 6\\ & y && 4 && 6 && 6 && 4 && 0 && \qquad \! \text{-}6\end{align*}
- \begin{align*}\ \\ & x && \text{-}3 && \text{-}2 && \text{-}1 && 0 && 1 && \qquad 2\\ & y && \! \text{-}27 && \text{-}8 && \text{-}1 && 0 && 1 && \qquad 8\\\end{align*}

Can the following data be modeled with an exponential function?

- \begin{align*}& x && 0 && 1 && 2 && 3 && 4 && \qquad 5\\ & y && \! \! \! 200 && \! \! \! 300 && \! \! \! \! 1800 && \! \! \! \! 8300 && \! \! \! \! \! 25, \! 800 && \quad 62, \! 700\end{align*}
- \begin{align*}\ \\ & x && 0 && 1 && 2 && 3 && 4 && \qquad 5\\ & y && \! \! \! 120 && \! \! \! 180 && \! \! \! 270 && \! \! \! 405 && \! \! \! \! 607.5 && \quad 911.25\\ \ \\\end{align*}
- \begin{align*}& x && 0 && 1 && 2 && 3 && 4 && \qquad 5\\ & y && \! \! \! \! 4000 && \! \! \! \! 2400 && \! \! \! \! 1440 && \! \! \! 864 && \! \! \! \! 518.4 && \quad 311.04\end{align*}

Determine whether the data is best represented by a quadratic, linear, or exponential function. Find the function that best models the data.

- \begin{align*}& x && 0 && 1 && 2 && 3 && 4\\ & y && \! \! \! 400 && \! \! \! 500 && \! \! 625 && \! \! \! \! 781.25 && \! \! \! \! \! \! \! \! 976.5625\end{align*}
- \begin{align*}\ \\ & x && \text{-}9 && \text{-}7 && \text{-}5 && \text{-}3 && \text{-}1 && \qquad 1\\ & y && \text{-}3 && \text{-}2 && \text{-}1 && 0 && 1 && \qquad 2\\ \ \\\end{align*}
- \begin{align*}& x && \text{-}3 && \! \text{-}2 && \text{-}1 && 0 && 1 && \qquad 2 && \qquad 3\\ & y && 14 && 4 && \text{-}2 && \! \text{-}4 && \! \text{-}2 && \qquad 4 && \qquad \! 14\end{align*}
- As a ball bounces up and down, the maximum height it reaches continually decreases. The table below shows the height of the bounce with regard to time.
- Using a graphing calculator, create a scatter plot of this data.
- Find the quadratic function of best fit.
- Draw the quadratic function of best fit on top of the scatter plot.
- Find the maximum height the ball reaches.
- Predict how high the ball is at 2.5 seconds.

Time (seconds) |
Height (inches) |
---|---|

2 | 2 |

2.2 | 16 |

2.4 | 24 |

2.6 | 33 |

2.8 | 38 |

3.0 | 42 |

3.2 | 36 |

3.4 | 30 |

3.6 | 28 |

3.8 | 14 |

4.0 | 6 |

- A chemist has a 250-gram sample of a radioactive material. She records the amount remaining in the sample every day for a week and obtains the following data.
- Draw a scatter plot of the data.
- Which function best suits the data: exponential, linear, or quadratic?
- Find the function of best fit and draw it through the scatter plot.
- Predict the amount of material present after 10 days.

Day |
Weight(grams) |
---|---|

0 | 250 |

1 | 208 |

2 | 158 |

3 | 130 |

4 | 102 |

5 | 80 |

6 | 65 |

7 | 50 |

- The following table show the pregnancy rate (per 1000) for U.S. women aged 15 – 19 (source: US Census Bureau). Make a scatter plot with the rate as the dependent variable and the number of years since 1990 as the independent variable. Find which model fits the data best. Use this model to predict the rate of teen pregnancy in the year 2010.

Year |
Rate of Pregnancy (per 1000) |
---|---|

1990 | 116.9 |

1991 | 115.3 |

1992 | 111.0 |

1993 | 108.0 |

1994 | 104.6 |

1995 | 99.6 |

1996 | 95.6 |

1997 | 91.4 |

1998 | 88.7 |

1999 | 85.7 |

2000 | 83.6 |

2001 | 79.5 |

2002 | 75.4 |

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.10.