Alice graphs the relationship between \( x \), the number of years an athlete has been playing basketball, and \( y \), their average number of baskets per game. Her graph goes through the points \( (8,8) \) and \( (4,5) \). Which equation represents the graph? A. \( y=\frac{3}{4} x+2 \) B. \( y=-\frac{3}{4} x-2 \) C. \( y=\frac{4}{3} x+\frac{31}{3} \) D. \( y=-\frac{4}{3} x-\frac{1}{3} \)

Let's first find the slope \( m \) between the two points \( (8,8) \) and \( (4,5) \). The slope formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 8}{4 - 8} = \frac{-3}{-4} = \frac{3}{4} \] Next, we can use the point-slope form of a line to find the equation. Using point \( (4, 5) \): \[ y - 5 = \frac{3}{4}(x - 4) \] Expand and rearrange: \[ y - 5 = \frac{3}{4}x - 3 \implies y = \frac{3}{4}x + 2 \] Thus, the equation that represents the graph is \( A. \) \( y=\frac{3}{4} x+2 \). For a bit of historical background, the concept of linear relationships has been pivotal in many scientific fields, helping in everything from predicting population growth to understanding economics. These basic principles can track observed behaviors and trends, making them invaluable in research. As for real-world applications, such linear equations often help coaches analyze players' performance over time. For instance, as an athlete gains more experience (years of play), their effectiveness can be predicted using the linear relationship shown, enabling tailored training and development plans geared towards enhancing their strengths, like shooting accuracy!