choosing the model that fits the data: linear, quadratic or exponential
Grade 11 Math Worksheet
Circle the best answer.
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 − 10x
- y = 50(0.64)^x
- y = 50(0.8)^x
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?
- Exponential: revenue grows fast at first, then levels off near the $96 peak
- Quadratic: revenue peaks at $10, the highest price tried
- 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
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?
- Exponential: each row is double the row before it, as the first rows show
- 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
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
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 = 3x + 1.4
- y = 3(1.4)^x
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?
- Sales and sunburns are barely related, because r is not exactly equal to 1
- Selling ice cream causes sunburns, because r = 0.88 is so close to 1 here
- Both rise on hot days, so r shows a link but not a cause
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 fits equally well across the x-values, because the residuals average about 0 across the eight points
- 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
Read this first
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.
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?
- Keep the whole data set and use the line, because r is simply what the data give, however odd the point looks
- 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
Table, x → y → ln y
1 → 6 → 1.79; 2 → 10 → 2.30; 3 → 17 → 2.83; 4 → 29 → 3.37; 5 → 50 → 3.91; 6 → 86 → 4.45. r for (x, y) is 0.935; r for (x, ln y) is 0.9999.
Noah fits y against x and then ln y against x. Technology gives the two r values shown. Which model does the data support?
- A line: r = 0.935 is already close to 1, so no change to the model is needed
- A power model: ln y against x is a line, so y is a power of x
- An exponential: ln y is almost linear in x, so y grows by a steady factor
- A quadratic: the points curve upward, so a parabola should fit them well
Fits y = 2,400(1.06)^x and y = 2,400(1.09)^x give two towns' library cardholders after x years. Which statement does Ari make comparing them?
- The second starts higher, because 1.09 is bigger than 1.06
- The second adds 3% more cardholders each year, a constant amount
- Both start at 2,400; the second grows 9% a year, the first 6%
- The first grows faster, because 1.06 is closer to 1