SAT Math: Scatterplots and Lines of Best Fit (Drill 3)

Question 1. Which of the following is true about the values of 3^x and 4x + 1 for x > 0?

• A) For all x > 0, it is true that 3^x < 4x + 1.
• B) For all x > 0, it is true that 3^x > 4x + 1.
• C) There is a constant c such that if 0 < x < c, then 3^x < 4x + 1, but if x > c, then 3^x > 4x + 1. ✓
• D) There is a constant c such that if 0 < x < c, then 3^x > 4x + 1, but if x > c, then 3^x < 4x + 1.

Explanation: At x = 0, 3⁰ = 1 and 4(0) + 1 = 1, so they are equal. At x = 1, 3¹ = 3 and 4(1) + 1 = 5, so 3^x < 4x + 1. At x = 2, 3² = 9 and 4(2) + 1 = 9, so they are equal again. At x = 3, 3³ = 27 and 4(3) + 1 = 13, so 3^x > 4x + 1. The exponential starts below the linear function, then overtakes it. There is a constant c = 2 where the crossover occurs: for 0 < x < 2, 3^x < 4x + 1, and for x > 2, 3^x > 4x + 1.

Question 2. Based on the line of best fit, what is the predicted revenue for month 8?

• A) $12,400
• B) $14,900 ✓
• C) $15,500
• D) $17,200

Explanation: Substitute x = 8: y = 1,550(8) + 2,500 = 12,400 + 2,500 = 14,900. Choice A (12,400) omits the y-intercept. Choice C (15,500) may result from a computation error. Note that this is an extrapolation beyond the data, so the prediction assumes the linear trend continues.

Question 3. A biologist models the concentration of a chemical in a lake using the equation C(t) = 120(0.85)^t, where C is the concentration in parts per million and t is the number of weeks since a cleanup began. Which of the following best describes what the model predicts?

• A) The concentration decreases by 0.85 parts per million each week.
• B) The concentration decreases by 85% each week.
• C) The concentration decreases by 15% each week. ✓
• D) The concentration increases by 15% each week.

Explanation: In the model C(t) = 120(0.85)^t, the base 0.85 represents the fraction retained each week. Since 0.85 = 1 − 0.15, the concentration retains 85% of its previous value each week, meaning it decreases by 15% per week. Choice A confuses the base with a constant decrease. Choice B reverses the interpretation, the concentration retains 85%, it doesn't lose 85%.

Question 4. A delivery company records the total fuel cost, in dollars, for trips of various distances. The line of best fit is y = 0.38x + 12.50, where x is the distance in miles and y is the fuel cost in dollars. Which of the following is the best interpretation of 0.38 in this context?

• A) The minimum fuel cost for any trip is $0.38.
• B) For each additional mile driven, the fuel cost increases by $0.38. ✓
• C) For each additional dollar spent on fuel, the distance increases by 0.38 miles.
• D) The average total fuel cost per trip is $0.38.

Explanation: The slope of a line of best fit represents the change in y for each one-unit increase in x. Here, x is distance in miles and y is fuel cost in dollars. So the slope of 0.38 means that for each additional mile driven, the predicted fuel cost increases by $0.38. Choice A describes a minimum, not a rate of change. Choice C reverses the x and y relationship.

Question 5. For which value of x is the residual (actual − predicted) the most negative?

• A) x = 1 ✓
• B) x = 2
• C) x = 3
• D) x = 5

Explanation: Calculate the residual (actual − predicted) for each choice. At x = 1: predicted = 5.6(1) + 58.5 = 64.1, residual = 62 − 64.1 = −2.1. At x = 2: predicted = 5.6(2) + 58.5 = 69.7, residual = 71 − 69.7 = +1.3. At x = 3: predicted = 5.6(3) + 58.5 = 75.3, residual = 74 − 75.3 = −1.3. At x = 5: predicted = 5.6(5) + 58.5 = 86.5, residual = 88 − 86.5 = +1.5. The most negative residual is −2.1 at x = 1, meaning the actual score was 2.1 points below the line of best fit. Choice C (x = 3) also has a negative residual (−1.3) but it is less negative than at x = 1.