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 for common difference

Find the differences between consecutive function - values.
\[s(2)-s(1)=9 - 7=2\]
\[s(3)-s(2)=11 - 9=2\]
\[s(4)-s(3)=13 - 11=2\]
Since the difference between consecutive \(y\) - values (values of \(s(x)\)) for a unit increase in \(x\) is constant (\(2\)), the function is linear.

Step2: Use the slope - intercept form \(y = mx + b\)

The slope \(m\) of a linear function is the common difference. Here \(m = 2\).
We know that when \(x = 1\), \(s(1)=7\). Substitute into \(s(x)=mx + b\) (where \(m = 2\) and \(x = 1\), \(s(x)=7\)):
\[7=2\times1 + b\]
\[7 = 2 + b\]
Solve for \(b\):
\[b=7 - 2=5\]

Step3: Write the equation of the linear function

The equation of the linear function \(s(x)\) is \(s(x)=2x + 5\).

Answer:

\(s(x)=2x + 5\)