choosing the model that fits the data: linear, quadratic or exponential
Grade 11 Math Worksheet - Challenge Me
Circle the best answer, or write your answer on the line.
Read this first
Table, week → cm
0 → 5; 1 → 6.5; 2 → 8.5; 3 → 11; 4 → 14.3; 5 → 18.6; 6 → 24.1. Technology fit: y = 5(1.3)^x, which predicts about 950 cm at week 20.
Jun fits a plant's heights for weeks 0 to 6 and predicts 950 cm, about 31 feet, at week 20. Is the prediction reasonable?
- No: plants cannot grow 30% in a week, so the model is wrong even for weeks 0 to 6
- No: week 20 is far past week 6, and a plant cannot keep growing 30% a week
- Yes: the model matches the seven weeks, so it will match week 20
Table, price in dollars → daily profit in dollars
4 → 18; 6 → 30; 8 → 34; 10 → 30; 12 → 18. Fit: P = −x² + 16x − 30, which gives −66 at x = 18.
Kira's fit for a bake sale's profit at prices $4 to $12 gives −$66 at a price of $18. Is that prediction reasonable?
- No: $18 is far past the prices tried, and −$66 shows the model failing
- Yes: a quadratic fits the five prices exactly, so the model should hold at $18 as well
- No: a stand cannot lose money, so a negative profit is impossible
- No: a quadratic is a poor model for profit, since profit is not a parabola
Table, day → views
1 → 120; 2 → 180; 3 → 270; 4 → 405; 5 → 608; 6 → 911; 7 → 1,367. Fit: y = 80(1.5)^x, which gives about 2,050 at day 8.
A video's views grew by about 1.5 times each day for seven days. Milo's fit predicts about 2,050 views at day 8. Is it reasonable?
- No: views cannot grow this fast for a whole week, so the model is too big
- Yes: day 8 is one step past a steady × 1.5 pattern, so 2,050 is fair
- No: a model cannot be trusted to predict anything past its data
Table, week → dollars
0 → 19.9; 1 → 21.8; 2 → 24.2; 3 → 25.2; 4 → 28.7; 5 → 30.0. Line: y = 2.06x + 19.81, r = 0.9924. Exponential: y = 20.06(1.0869)^x, r = 0.9927. At week 12: line 44.6, exponential 54.6.
Tala fits both models to weeks 0 to 5. Their predictions for week 12 are 44.6 and 54.6. Which statement does the data support best?
- The line's 44.6, because the smaller prediction is the safer choice to report to the team
- Neither is settled: both fit equally well, so how the process works must decide
- The exponential's 54.6, because its r is slightly higher than the line's
Table, day → cm
10 → 12.1; 15 → 28.2; 20 → 43.8; 25 → 60.1; 30 → 76.0. Fit: y = 3.2x − 20.
Luz fits a vine's height for days 10 to 30 and says the −20 in the fit means the vine started 20 cm below ground. Is that a good reading?
- No: a height cannot be negative, so the fit must have an arithmetic error
- No: day 0 is outside the data, so −20 is where the line meets the axis
- No: the intercept of a growing vine is zero
- Yes: the y-intercept of a fitted line is the starting value
Table, hour → thousand cells
0 → 64; 2 → 96; 4 → 144; 6 → 216. Linear: y = 25.2x + 54.4. Exponential: y = 64(1.2247)^x. Quadratic: y = 2.5x² + 10.2x + 64.4.
Noah gets three fits from technology for the cells in a lab dish. Using the one the table supports, what is y at x = 9, to the nearest whole number?
Table, year → dollars
0 → 3,000; 3 → 2,400; 6 → 1,920; 9 → 1,536. Linear: y = −162.4x + 2,944.8. Exponential: y = 3,000(0.9283)^x. Quadratic: y = 6x² − 216.4x + 2,998.8.
Bo gets three fits for a car's value. Using the one the table supports, after how many years does the value reach $1,000, to the nearest tenth?
Table, second → feet
0 → 4; 1 → 36; 2 → 36. Linear: y = 16x + 9.33. Quadratic: y = −16x² + 48x + 4.
Esme throws a ball from a 4-foot-high hand and gets two fits. Using the one the table supports, when does the ball reach the ground, to the nearest tenth of a second?
Table, price in dollars → weekly profit in dollars
1 → 7; 3 → 43; 4 → 52; 8 → 28; 9 → 7. Linear: y = −1.17x + 33.27. Quadratic: y = −3x² + 30x − 20.
Ari runs a bike-repair stand and gets two fits. Using the one the table supports, what is the greatest weekly profit the model gives, in dollars?
Table, week → leaves
1 → 7; 2 → 15; 3 → 31; 4 → 66; 5 → 135. Fit of y on x: y = 30.7x − 41.3. Fit of ln y on x: ln y = 0.74x + 1.216.
Pia counts the leaves on a vine each week and gets two fits from technology. Using the one the table supports, what is y at x = 7, to the nearest whole number?
