Worksheet

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

Grade 11 Math Worksheet

NameDate

Circle the best answer.

  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
    • Quadratic: it peaks near $7, halfway between the equal $4 and $10 rows
    • Exponential: revenue grows fast at first, then levels off near the $96 peak
    • Quadratic: revenue peaks at $10, the highest price tried

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

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

  4. 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 = 3x + 1.4
    • y = 3(1.5)^x
    • y = 3(1.4)^x

  5. 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.

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

  7. 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

  8. 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 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
    • The line is centered, but the errors grow with x, so large-x predictions are less reliable

  9. 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.

  10. 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?

    • 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
    • Check whether (9, 3) is a mistake or a real case, then compare fits with and without it

  11. 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.

  12. 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
    • An exponential: ln y is almost linear in x, so y grows by a steady factor
    • A power model: ln y against x is a line, so y is a power of x
    • A quadratic: the points curve upward, so a parabola should fit them well

  13. Technology fits y = 800(0.75)^(x/2) to a phone's resale value in dollars after x years. Which description does Luz choose?

    • Starts at 800 and falls 12.5% each year, half of the 25% over 2 years
    • Starts at 800 and falls about 13.4% each year, 25% every 2 years
    • Starts at 0.75 and is multiplied by 800 every 2 years
    • Starts at 800 and falls 25% each year, so about 600 are left after 1 year
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choosing the model that fits the data: linear, quadratic or exponential

Answer Key - Grade 11 Math Worksheet

Circle the best answer.

  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
    • Quadratic: it peaks near $7, halfway between the equal $4 and $10 rows
    • Exponential: revenue grows fast at first, then levels off near the $96 peak
    • Quadratic: revenue peaks at $10, the highest price tried

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

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

  4. 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 = 3x + 1.4
    • y = 3(1.5)^x
    • y = 3(1.4)^x

  5. 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.

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

  7. 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

  8. 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 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
    • The line is centered, but the errors grow with x, so large-x predictions are less reliable

  9. 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.

  10. 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?

    • 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
    • Check whether (9, 3) is a mistake or a real case, then compare fits with and without it

  11. 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.

  12. 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
    • An exponential: ln y is almost linear in x, so y grows by a steady factor
    • A power model: ln y against x is a line, so y is a power of x
    • A quadratic: the points curve upward, so a parabola should fit them well

  13. Technology fits y = 800(0.75)^(x/2) to a phone's resale value in dollars after x years. Which description does Luz choose?

    • Starts at 800 and falls 12.5% each year, half of the 25% over 2 years
    • Starts at 800 and falls about 13.4% each year, 25% every 2 years
    • Starts at 0.75 and is multiplied by 800 every 2 years
    • Starts at 800 and falls 25% each year, so about 600 are left after 1 year