choosing the model that fits the data: linear, quadratic or exponential

Grade 11 Math Worksheet

Circle the best answer.

1. 1.

   2 → 48; 4 → 80; 8 → 96; 10 → 80. Bo's lemonade stand tried four prices and recorded the daily revenue. Which model fits, and where does revenue peak?

   • Quadratic: revenue peaks at $8, because 96 is the largest value in the table
   • Exponential: revenue grows fast at first, then levels off near the $96 peak
   • Quadratic: revenue peaks at $10, the highest price tried
   • Quadratic: it peaks near $7, halfway between the equal $4 and $10 rows
2. 2.

   2 → 18; 5 → 144. Zee knows a lab dish grows by a constant factor each hour, and has two readings. Which equation y = a(b)^x passes through both?

   • y = 18(8)^x
   • y = 18(2)^x
   • y = 4.5(2)^x
3. 3.

   0 → 180; 10 → 125; 20 → 97.5; 30 → 83.75. The room is 70 °F. Pia records a cup of cocoa cooling in a 70 °F room. The temperature never goes below 70. Which model fits the table?

   • The temperature halves every 10 minutes
   • Linear: the temperature falls by about 55 degrees every 10 minutes
   • The temperature minus 70 halves every 10 minutes
   • Quadratic: the drops 55, 27.5 and 13.75 shrink by a steady amount

4. Read this first

   Table, week → cm

   1 → 2.1; 2 → 2.6; 3 → 3.9; 4 → 6.2; 5 → 9.1; 6 → 12.8; 7 → 17.0. Line: y = 2.51x − 2.37, r = 0.97. Residuals: +2.0, 0.0, −1.3, −1.5, −1.1, +0.1, +1.8.

5. 4.

   Luz fits a line to a plant's height by week. Technology gives r = 0.97, but the residuals form a U. Which conclusion is best?

   • The line fits well: r = 0.97 is close to 1, so the residuals do not matter
   • The line is a poor model: a high r can hide a curve, so a quadratic is better
   • The line fits: the residuals are small, from −1.5 to +2.0
6. 5.

   Kira fits a line and an exponential to the same six points. Both have r above 0.96. Which model should Kira use?

   • The exponential: its residuals are small and scattered; the line's form a U
   • Either one: both have r above 0.96, so they are equally good for these points
   • The line: it has fewer parts, and its residuals are about as small as the curve's
7. 6.

   Milo fits a line to eight points. The residuals alternate in sign and the later ones are much larger. Which conclusion is best?

   • The line is centered, but the errors grow with x, so large-x predictions are less reliable
   • The line fits equally well across the x-values, because the residuals average about 0 across the eight points
   • A curve fits better, because the residuals alternate above and below 0 as x grows
   • A curve fits better, because the line's r is just 0.93

8. Points

   (1, 2.4), (2, 3.1), (3, 4.9), (4, 5.2), (5, 6.8), (6, 7.1), (7, 8.9), (9, 3). With all eight, r = 0.43. Without (9, 3), r = 0.99.

9. 7.

   A line through eight points has r = 0.43. Without the point (9, 3), the other seven give r = 0.99. What should Tala do first?

   • Check whether (9, 3) is a mistake or a real case, then compare fits with and without it
   • Delete the point at once, because removing it raises r from 0.43 to 0.99
   • Keep the whole data set and use the line, because r is simply what the data give, however odd the point looks
10. 8.

    Pia fits a line to six points and gets the residuals shown, a U. Just beyond x = 6, will the line likely be too high or too low?

    • Too low: the data curve upward, so the line falls further below them
    • Too high: the line goes through the middle of the data, so it overestimates
    • Right on target: the residuals average 0, so the line is exact
11. 9.

    Esme's fit y = 2.4x + 15 gives a vine's height in cm after x weeks. Esme rewrites it with x in days. Which equation matches?

    • y = 16.8x + 15, because there are 7 days in a week
    • y = 0.34x + 15, because only the slope changes
    • y = 0.34x + 2.1, because both numbers are divided by 7
12. 10.

    Technology fits y = 900(0.9)^x to the grams of ice left in a cooler after x hours. Which description does Zee choose?

    • Starts at 900 and falls 10% each hour, so it is half gone after 5 hours
    • Starts at 900 and loses 90 grams each hour, so it is gone after exactly 10 hours
    • Starts at 900 and falls 10% each hour, so it is half gone between hours 6 and 7
    • Starts at 900 and falls 90% each hour, so it is nearly gone after an hour