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

Grade 11 Math Worksheet

Circle the best answer.

1. 1.

   0 → 50; 1 → 40; 3 → 25.6. Tala tracks a vehicle's value in thousands of dollars at years 0, 1 and 3. Which equation fits all three rows?

   • y = 50(0.8)^x
   • y = 50(0.64)^x
   • y = 50 − 10x
2. 2.

   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 $10, the highest price tried
   • Exponential: revenue grows fast at first, then levels off near the $96 peak
   • Quadratic: revenue peaks at $8, because 96 is the largest value in the table
   • Quadratic: it peaks near $7, halfway between the equal $4 and $10 rows
3. 3.

   0 → 1; 1 → 2; 2 → 4; 3 → 7; 4 → 11; 5 → 16. Ari notices the first rows of a growth table double: 1, 2, 4. Here is the whole table. Which model fits every row?

   • Linear: y rises by about 3 per row, 15 over 5 rows
   • Quadratic: the adds 1, 2, 3, 4 and 5 grow by a steady 1
   • Exponential: each row is double the row before it, as the first rows show
4. 4.

   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(2)^x
   • y = 18(8)^x
   • y = 4.5(2)^x
5. 5.

   1 → 4.3; 2 → 5.7; 3 → 8.4; 4 → 11.2; 5 → 16.4. Ren has a video's weekly views for weeks 1 to 5. Which equation is within 1 of every one of the five values?

   • y = 3(1.5)^x
   • y = 3(1.4)^x
   • y = 3x + 1.4

6. 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.

7. 6.

   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 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
   • The line fits well: r = 0.97 is close to 1, so the residuals do not matter
8. 7.

   Over 20 summer days, a town's ice-cream sales and sunburn cases have r = 0.88. Which conclusion does Jun say the data support?

   • Selling ice cream causes sunburns, because r = 0.88 is so close to 1 here
   • Sales and sunburns are barely related, because r is not exactly equal to 1
   • Both rise on hot days, so r shows a link but not a cause
9. 8.

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

   • Either one: both have r above 0.96, so they are equally good for these points
   • The exponential: its residuals are small and scattered; the line's form a U
   • The line: it has fewer parts, and its residuals are about as small as the curve's
10. 9.

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

    • A curve fits better, because the residuals alternate above and below 0 as x grows
    • The line is centered, but the errors grow with x, so large-x predictions are less reliable
    • A curve fits better, because the line's r is just 0.93
    • The line fits equally well across the x-values, because the residuals average about 0 across the eight points
11. 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 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
    • Starts at 900 and falls 10% each hour, so it is half gone after 5 hours