Question

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?
show your work here

b
○ $y = \frac{3}{4}x + 2$ ○ $y = -\frac{3}{4}x - 2$
○ $y = \frac{3}{4}x + \frac{11}{4}$ ○ $y = -\frac{3}{4}x - \frac{1}{4}$

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using the points \((8, 8)\) and \((4, 5)\), we have \( m=\frac{5 - 8}{4 - 8}=\frac{-3}{-4}=\frac{3}{4} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \((8, 8)\) and \( m = \frac{3}{4} \), we get \( y - 8=\frac{3}{4}(x - 8) \).

Step3: Simplify the equation

Expand the right - hand side: \( y - 8=\frac{3}{4}x-6 \). Then add 8 to both sides: \( y=\frac{3}{4}x - 6 + 8=\frac{3}{4}x+2 \). We can also check with the other point \((4,5)\). Substitute \( x = 4 \) into \( y=\frac{3}{4}x + 2 \), we get \( y=\frac{3}{4}\times4+2=3 + 2=5 \), which matches the point \((4,5)\).

Answer:

\( y=\frac{3}{4}x + 2 \) (the first option in the given choices)