Names ______
Modeling Using Quadratic Functions
Activity
FALLING WATER: Water drains from a hole made in a 2-liter bottle. The table shows the level of the water y measured in centimeters from the bottom of the bottle after x seconds. Find and graph a linear regression equation and a quadratic regression equation.
Determine which equation is a better fit for the data.
Step 1 Find a linear regression equation.
(Remember, you are entering and working with statistics. The STAT button is used most of the time)
Enter the times in L1 and the water levels in L2. Remember to turn on the STAT PLOT when you’re graphing the scatter plot. Then find a linear regression equation (menu STAT – CALC). Graph a scatter plot and the regression equation on the same screen. If you don’t remember how to copy the regression equation into the Y= screen, see Step 2 for the keystrokes. It should look like the example to the right when you’re finished. {Notice the change in window parameters. [0, 260] refers to xmax and xmin and you can see the change in scale.}
1. What linear regression equation does the calculator produce from this data?
Round to the nearest thousandth for the slope and tenth for the y-intercept.
After you have both the scatter plot and regression line on your screen, analyze the prediction equation by using either the TRACE or CALC-VALUE feature. You may have to adjust your window.
2. Is this line going to be useful for predicting future water levels? If not, at approximately what point does the regression equation become invalid? Why?
Step 2 Find a quadratic regression equation.
Find the quadratic regression equation (it’s under the same menu as the linear regression equation). Then copy the equation to the Y=list and graph.
Keystrokes for copying:
3. What quadratic regression equation does the calculator produce from this data?
Round a, b and c to the nearest tenth.
4. Analyze the curve and write down which prediction equation (linear or quadratic) fits the data best. Does the quadratic regression equation become invalid? Test future values and tell why.
For Exercises 1–4, use the graph of the braking distances for dry pavement. Put your answers on a separate sheet of paper and staple both sheets together.
1. Find and graph a linear regression equation and a quadratic regression equation for the data. Determine which equation is a better fit for the data.
2. Use the CALC menu with each regression equation to estimate the braking distance at speeds of 100 and 150 miles per hour.
3. How do the estimates found in Exercise 2 compare?
4. How might choosing a regression equation that does not fit the data well affect predictions made by using the equation?