Question

two points defining a linear function are shown in the table below.

x	y
-10	-12

what is the slope of the function?
options: $\frac{2}{3}$, $\frac{3}{2}$, 2, 6

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify points

Let \((x_1, y_1)=(-14, -18)\) and \((x_2, y_2)=(-10, -12)\).

Step3: Substitute into formula

Calculate \( y_2 - y_1=-12 - (-18)=-12 + 18 = 6 \).
Calculate \( x_2 - x_1=-10 - (-14)=-10 + 14 = 4 \).
Then \( m=\frac{6}{4}=\frac{3}{2} \)? Wait, no, wait: Wait, \( x_2 - x_1=-10 - (-14)=4 \), \( y_2 - y_1=-12 - (-18)=6 \)? Wait, no, wait, let's recalculate: \( y_2 - y_1=-12 - (-18)=6 \), \( x_2 - x_1=-10 - (-14)=4 \)? Wait, no, wait, the two points are \((-14, -18)\) and \((-10, -12)\). So \( x_1=-14, y_1=-18 \); \( x_2=-10, y_2=-12 \). Then \( y_2 - y_1=-12 - (-18)=6 \), \( x_2 - x_1=-10 - (-14)=4 \)? Wait, no, \( -10 - (-14)=4 \), yes. Then slope is \( \frac{6}{4}=\frac{3}{2} \)? Wait, but let's check again. Wait, maybe I mixed up. Wait, \( y_2 - y_1=-12 - (-18)=6 \), \( x_2 - x_1=-10 - (-14)=4 \), so \( \frac{6}{4}=\frac{3}{2} \). Wait, but let's do it again. Alternatively, \( (y_1 - y_2)/(x_1 - x_2)=(-18 - (-12))/(-14 - (-10))=(-6)/(-4)=\frac{3}{2} \). Yes, that's correct. Wait, but wait, the options have \( \frac{3}{2} \) as an option. Wait, but let's check the calculation again. Wait, \( x \) changes from -14 to -10, which is an increase of 4. \( y \) changes from -18 to -12, which is an increase of 6. So slope is rise over run, which is 6/4 = 3/2. Yes.

Answer:

\(\frac{3}{2}\) (corresponding to the option with \(\frac{3}{2}\))