We are given a table that shows a familiar pattern from geometry.
x
1
2
3
4
5
y
Ï€
2Ï€
3Ï€
4Ï€
5Ï€
From the given figure, we can see that x represents the diameter of the given circle. To find the meaning of y, recall that functions written in slope-intercept form follow a specific format.
y= mx+ b
In this form, m is the slope and b is the y-intercept. We need to identify these values using the given table. Let's start by substituting the points (1,Ï€) and (2,2Ï€), into the slope formula and calculating m.
We calculated that the slope of the given equation is π. Now, remember that the y-intercept is the y-value where the line crosses the y-axis. It occurs when x=0. Since the slope is equal to π, we need to subtract 1 unit to x-coordinate and π units to y-coordinate of the first given point to obtain the previous point.
(x_1,y_1)
(x-1,y-2)
(x_2,y_2)
(1,Ï€)
(1- 1,π- π)
(0,0)
Notice that we obtain a point where x=0. This means that the function intercepts the y-axis at (0, 0). Therefore, the value of b is 0. Now that we have the slope and the y-intercept, we can write a function for the line that passes through the given points.
y= πx+ 0 → y=π x
Now, recall the formula for the circumference of a circle.
C= π d
Here, d represents the diameter of the circle and C the circumference. In our case, the diameter is equal to x. Let's substitute this value into the formula.
We get the same form as the function obtained with the given data. Therefore, x represents the diameter and y the circumference of the circle.
Remember that a linear function can be written in the following form.
y= mx+ b
Let's compare this function with the obtained in Part A.
y=π x → y= πx+ 0
This means that our function is a linear function.
