# College Math – Quadratic and Exponential regression

**Please be sure to do BOTH the quadratic AND exponential regression projects.**

**Quadratic Model**

**QR–****â€****4: Input Time of Day (hour) **14.5

**QR–****â€****6: Target Outdoor Temperature **53.0

**Exponential Model**

**ER–****â€****4: Input Elapsed Time (minutes) **102

**ER–****â€****5: Target Coffee Temperature **96

**Tasks for Quadratic Regression Model (QR)**

**(QR-1)**Plot the points (*x*,*y*) to obtain a**scatterplot**. Note that the trend is definitely non-linear. Use an appropriate scale on the horizontal and vertical axes and be sure to label carefully.

**(QR-2)**Find the**quadratic polynomial of best fit**and**graph**it on the scatterplot. State the**formula**for the quadratic polynomial.

**(QR-3)**Find and state the value of**r**** ^{2}**, the coefficient of determination. Discuss your findings. (

*r*

^{2}is calculated using a different formula than for linear regression. However, just as in the linear case, the closer

*r*

^{2}is to 1, the better the fit.Just work with

**r****, not**

^{2}*r*.) Is a parabola a good curve to fit to this data?

**(QR-4)**Use the quadratic polynomial to make an outdoor temperature**estimate**. Each class member will compute a temperature estimate for a different time of day assigned by your instructor. Be sure to use the quadratic regression model to make the estimate (not the values in the data table). State your results clearly — the time of day and the corresponding outdoor temperature estimate.

**(QR-5)**Using algebraic techniques we have learned, find the**maximum temperature****predicted by the quadratic model **and find the**time when it occurred.** Report the time to the nearest quarter hour (i.e., __:00 or __:15 or __:30 or __:45). (For instance, a time of 18.25 hours is reported as 6:15 pm.) Report the maximum temperature to the nearest tenth of a degree. **Show work.**

**(QR-6)**Use the quadratic polynomial together with algebra to**estimate the time(s) of day when the outdoor temperature is a specific target temperature. **Each class member will work with a different target temperature, assigned by your instructor. Report the time(s) to the nearest quarter hour. Be sure to use the quadratic model to make the time estimates (not values in the data table). **Show work.** State your results clearly — the target temperature and the associated time(s).**Show work.**

** **

**Tasks for Exponential Regression Model (ER)**

**(ER-1)**Plot the points (*x*,*y*) in the second table (Table 2) to obtain a**scatterplot**. Note that the trend is definitely non-linear. Use an appropriate scale on the horizontal and vertical axes and be sure to label carefully.

**(ER-2)**Find the**exponential function of best fit**and**graph**it on the scatterplot. State the**formula**for the exponential function. It should have the form*y*=*A**e*^{–bx} where software has provided you with the numerical values for*A*and*b*.

**(ER-3)**Find and state the value of**r**** ^{2}**, the coefficient of determination. Discuss your findings.(

*r*

^{2}is calculated using a different formula than for linear regression. However, just as in the linear case, the closer

*r*

^{2}is to 1, the better the fit.) Is an exponential curve a good curve to fit to this data?

**(ER-4)**Use the exponential function to make a**coffee temperature****estimate**. Each class member will compute a temperature estimate for a different elapsed time *x* assigned by your instructor. Substitute your*x*value into your exponential function to get*y*, the corresponding temperature difference between the coffee temperature and the room temperature. Since*y*=*T*– 69, we have coffee temperature*T*=*y*+ 69. Take your*y*estimate and add 69 degrees to get the coffee temperature estimate. State your results clearly — the elapsed time and the corresponding estimate of the coffee temperature.

**(ER-5)**Use the exponential function together with algebra to**estimate the elapsed time when the coffee arrived at a particular target temperature**. Report the elapsed time to the nearest tenth of a minute. Each class member will work with a different target coffee temperature *T* assigned by your instructor.

Given your target temperature *T*, then*y*=*T*– 69 is your target temperature *difference* between the coffee and room temperatures. Use your exponential model *y*=*A**e*^{–bx}. Substitute your target temperature difference for *y* and solve the equation*y*=*A**e*^{–bx}for elapsed time *x. ***Show algebraic work**in solving your equation. State your results clearly — your target temperature and the estimated elapsed time, to the nearest tenth of a minute.

For instance, if the target coffee temperature*T*= 150 degrees, then *y* = 150 – 69 = 81 degrees is the temperature *difference* between the coffee and the room, what we are calling*y*. So, for this particular target coffee temperature of 150 degrees, the goal is finding how long it took for the temperature difference*y*to arrive at 81 degrees; that is, solving the equation 81 =*A**e*^{–bx}for*x*.