Question

1. the table represents some points on the graph of a linear function.

x	y
-4	10
-2	7
6	-5

which function represents this relationship?

• $y = -\frac{3}{2}x + 4$
• $y = -\frac{3}{2}x - 14$
• $y = -\frac{2}{3}x - 9$
• $y = -\frac{2}{3}x - 1$

Explanation:

Step1: Calculate the slope

Use two points \((-4, 10)\) and \((-2, 7)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). So \(m=\frac{7 - 10}{-2-(-4)}=\frac{-3}{2}=-\frac{3}{2}\).

Step2: Find the equation

Use the point - slope form \(y - y_1=m(x - x_1)\). Let's use the point \((-4,10)\) and \(m =-\frac{3}{2}\). Then \(y - 10=-\frac{3}{2}(x + 4)\). Expand it: \(y-10=-\frac{3}{2}x-6\). Add 10 to both sides: \(y=-\frac{3}{2}x + 4\). We can also check by plugging in the points into the equation. For \(x=-4\), \(y =-\frac{3}{2}\times(-4)+4 = 6 + 4=10\) (matches). For \(x=-2\), \(y=-\frac{3}{2}\times(-2)+4=3 + 4 = 7\) (matches). For \(x = 6\), \(y=-\frac{3}{2}\times6+4=-9 + 4=-5\) (matches).

Answer:

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