Question

at the left you can see 4 clusters of circles with the number of circles in each cluster written below it. determine if the function s(x) that maps the cluster to its number of circles is a linear function. if it is linear, find the equation of the function s(x). s(1)=7 s(2)=9 s(3)=11 s(4)=13 s(x) is not a linear function s(x)=

Explanation:

Step1: Check the slope

The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $x_1 = 1,y_1 = 7,x_2=2,y_2 = 9$. Then $m=\frac{9 - 7}{2 - 1}=2$. Let $x_2 = 2,y_2 = 9,x_3=3,y_3 = 11$, then $m=\frac{11 - 9}{3 - 2}=2$. Let $x_3 = 3,y_3 = 11,x_4=4,y_4 = 13$, then $m=\frac{13 - 11}{4 - 3}=2$. Since the slope between any two - consecutive points is constant ($m = 2$), it is a linear function.

Step2: Use the point - slope form

The point - slope form of a linear equation is $y - y_1=m(x - x_1)$. Using the point $(x_1,y_1)=(1,7)$ and $m = 2$, we have $y-7=2(x - 1)$.

Step3: Simplify the equation

Expand the right - hand side: $y-7=2x-2$. Add 7 to both sides of the equation: $y=2x + 5$. So $s(x)=2x + 5$.

Answer:

$s(x)=2x + 5$