Question

3 jacob and his classmate are competing to see who can earn more pbis points. jacobs pbis points can be represented by the table below. weeks, x pbis points, y 2 15 4 39 5 51 8 87 jacobs classmates pbis points can be represented as y = 14x + 10 which of the following statements is true? a jacob has more pbis points to begin with than his classmate. b jacobs classmate earns more pbis points per week than jacob. c jacobs classmate starts out with fewer pbis points than jacob. d jacob has more pbis points at week 8 than his classmate.

Explanation:

Step1: Find Jacob's slope (points per week)

Use two points, e.g., (2,15) and (4,39). Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{39 - 15}{4 - 2} = \frac{24}{2} = 12 \).

Step2: Find Jacob's y-intercept (initial points)

Use point (2,15) and slope 12. \( y = mx + b \Rightarrow 15 = 12(2) + b \Rightarrow 15 = 24 + b \Rightarrow b = -9 \).

Step3: Analyze classmate's equation \( y = 14x + 10 \)

• Slope (points per week): 14.
• Y-intercept (initial points): 10.

Step4: Evaluate options

• Option A: Jacob's initial points: -9; Classmate: 10. So classmate has more initially. A is false.
• Option B: Jacob's slope: 12; Classmate's slope:14. 14 > 12, so classmate earns more per week. B is true.
• Option C: Classmate's initial:10; Jacob's: -9. Classmate has more initially. C is false.
• Option D: Jacob at week 8: \( y = 12(8) - 9 = 96 - 9 = 87 \). Classmate at week 8: \( y = 14(8) + 10 = 112 + 10 = 122 \). 87 < 122, so classmate has more. D is false.

Answer:

B. Jacob's classmate earns more PBIS points per week than Jacob.