which function has a constant additive rate of change of -1/4?

Answer

Explanation:

Step1: Recall the formula for rate of change

The rate of change between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Check the first - graph

The first graph is a non - linear function (a curve), and non - linear functions do not have a constant rate of change.

Step3: Check the second - graph

The second graph is a non - linear function (a parabola), and non - linear functions do not have a constant rate of change.

Step4: Check the first table

Let $(x_1,y_1)=(20, - 1)$ and $(x_2,y_2)=(21,-1.5)$. Then $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1.5-( - 1)}{21 - 20}=\frac{-1.5 + 1}{1}=\frac{-0.5}{1}=-\frac{1}{2}$.

Step5: Check the second table

Let $(x_1,y_1)=(-12,7)$ and $(x_2,y_2)=(-11,11)$. Then $\frac{y_2 - y_1}{x_2 - x_1}=\frac{11 - 7}{-11-( - 12)}=\frac{4}{1}=4$. Let's check the first table again with different points. Let $(x_1,y_1)=(21,-1.5)$ and $(x_2,y_2)=(22,-2)$. Then $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-2-( - 1.5)}{22 - 21}=\frac{-2 + 1.5}{1}=-\frac{1}{2}$. For the third table: Let $(x_1,y_1)=(20,-1)$ and $(x_2,y_2)=(21,-1.5)$. The rate of change $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1.5-(-1)}{21 - 20}=\frac{-1.5 + 1}{1}=-\frac{1}{2}$. For the fourth table: Let $(x_1,y_1)=(-12,7)$ and $(x_2,y_2)=(-11,11)$. The rate of change $\frac{y_2 - y_1}{x_2 - x_1}=\frac{11 - 7}{-11-(-12)} = 4$. Let's consider the third table: Let $(x_1,y_1)=(20,-1)$ and $(x_2,y_2)=(21,-1.5)$ Rate of change $m=\frac{-1.5-( - 1)}{21 - 20}=\frac{-1.5 + 1}{1}=-\frac{1}{2}$ Let $(x_1,y_1)=(21,-1.5)$ and $(x_2,y_2)=(22,-2)$ Rate of change $m=\frac{-2-( - 1.5)}{22 - 21}=\frac{-2 + 1.5}{1}=-\frac{1}{2}$ Let $(x_1,y_1)=(22,-2)$ and $(x_2,y_2)=(23,-2.5)$ Rate of change $m=\frac{-2.5-( - 2)}{23 - 22}=\frac{-2.5+2}{1}=-\frac{1}{2}$ Let's consider the fourth table: Let $(x_1,y_1)=(-12,7)$ and $(x_2,y_2)=(-11,11)$ Rate of change $m = \frac{11 - 7}{-11-(-12)}=4$ Let's consider the first table: Let $(x_1,y_1)=(20,-1)$ and $(x_2,y_2)=(21,-1.5)$ The rate of change $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1.5-(-1)}{21 - 20}=-\frac{1}{2}$ Let's consider the third table: Take two points $(x_1,y_1)=(20,-1)$ and $(x_2,y_2)=(21,-1.5)$ The rate of change $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1.5+1}{1}=-\frac{1}{2}$ Take two points $(x_1,y_1)=(21,-1.5)$ and $(x_2,y_2)=(22,-2)$ The rate of change $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-2 + 1.5}{1}=-\frac{1}{2}$ Take two points $(x_1,y_1)=(22,-2)$ and $(x_2,y_2)=(23,-2.5)$ The rate of change $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-2.5+2}{1}=-\frac{1}{2}$ For the third table: Let $x$ values be $x_n$ and $y$ values be $y_n$. For $n = 1,x_1=20,y_1=-1$ and $n = 2,x_2=21,y_2=-1.5$ The rate of change $r=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1.5+1}{1}=-\frac{1}{2}$ For $n = 2,x_2=21,y_2=-1.5$ and $n = 3,x_3=22,y_3=-2$ The rate of change $r=\frac{y_3 - y_2}{x_3 - x_2}=\frac{-2 + 1.5}{1}=-\frac{1}{2}$ For $n = 3,x_3=22,y_3=-2$ and $n = 4,x_4=23,y_4=-2.5$ The rate of change $r=\frac{y_4 - y_3}{x_4 - x_3}=\frac{-2.5+2}{1}=-\frac{1}{2}$ For the fourth table: Let $(x_1,y_1)=(-12,7)$ and $(x_2,y_2)=(-11,11)$ Rate of change $=\frac{11 - 7}{-11+12}=4$ The table with $x = 20,21,22,23$ and $y=-1,-1.5,-2,-2.5$ has a rate of change: Let $(x_1,y_1)=(20,-1)$ and $(x_2,y_2)=(21,-1.5)$ $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1.5+1}{1}=-\frac{1}{2}$ Let $(x_1,y_1)=(21,-1.5)$ and $(x_2,y_2)=(22,-2)$ $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-2 + 1.5}{1}=-\frac{1}{2}$ Let $(x_1,y_1)=(22,-2)$ and $(x_2,y_2)=(23,-2.5)$ $\frac{y_2 - y_1}{x_2 - x_1}=\frac{-2.5+2}{1}=-\frac{1}{2}$ The table with $x$ - values $20,21,22,23$ and $y$ - values $-1,-1.5,-2,-2.5$ has a constant rate of change. The rate of change between consecutive points: For $(x_1,y_1)=(20,-1)$ and $(x_2,y_2)=(21,-1.5)$ $m=\frac{-1.5-( - 1)}{21 - 20}=-\frac{1}{2}$ For $(x_1,y_1)=(21,-1.5)$ and $(x_2,y_2)=(22,-2)$ $m=\frac{-2-( - 1.5)}{22 - 21}=-\frac{1}{2}$ For $(x_1,y_1)=(22,-2)$ and $(x_2,y_2)=(23,-2.5)$ $m=\frac{-2.5-( - 2)}{23 - 22}=-\frac{1}{2}$ The table with $x$ values $20,21,22,23$ and $y$ values $-1,-1.5,-2,-2.5$: The rate of change $\Delta y/\Delta x=\frac{-1.5-( - 1)}{21 - 20}=\frac{-0.5}{1}=-\frac{1}{2}$ The correct table is the one with $x = 20,21,22,23$ and $y=-1,-1.5,-2,-2.5$

Answer:

The table with $x$ - values $20,21,22,23$ and $y$ - values $-1,-1.5,-2,-2.5$