Question

given the following relationships we know, list the relationships in order from least to greatest based on the constant of proportionality. use the table, graph and equation to show them to this applicable position with the smallest constant of proportionality at the top.
a.

x	1	3	4	12
y	6	2	1	0.5

b.

c.
y = \frac{24}{x}

Explanation:

Step1: Find constant for Table A

Inverse variation: $xy = k$.
For $x=1,y=6$: $k_1 = 1\times6=6$
For $x=3,y=2$: $k_2 = 3\times2=6$
For $x=6,y=1$: $k_3 = 6\times1=6$
For $x=12,y=0.5$: $k_4 = 12\times0.5=6$
So $k_A=6$

Step2: Find constant for Graph B

Inverse variation: $xy = k$.
Use point $(1,2.5)$: $k_B = 1\times2.5=2.5$
Verify with $(5,0.5)$: $5\times0.5=2.5$, so $k_B=2.5$

Step3: Find constant for Equation C

Equation: $y=\frac{25}{x}$, so $k_C=25$

Step4: Order constants from least to greatest

Compare $2.5, 6, 25$: $2.5 < 6 < 25$

Answer:

Graph B ($k=2.5$), Table A ($k=6$), Equation C ($k=25$)