Question

Ule

The table below models a particular physical situation.
X -10 -3 3 10
-8 -9 -5 2
y
Find the piecewise linear equation that models the data above. Round to three decimal places if needed.
y =
x +
x+
x+
-10 ≤x≤-3
-3 < x≤3
3 < x≤ 10

Answer 1

To find the piecewise linear equation that models the given data, you need to determine the equations for each segment of the data where the x-values fall within certain ranges. Let's calculate the equations for each segment:

For -10 ≤ x ≤ -3:
The two points in this segment are (-10, -8) and (-3, -9). You can find the slope (m) using the formula m = (y2 - y1) / (x2 - x1):
m = (-9 - (-8)) / (-3 - (-10)) = (-9 + 8) / (-3 + 10) = -1 / 7

Now, you can use the point-slope form of a line to find the equation for this segment:
y - (-8) = (-1/7)(x - (-10))
y + 8 = (-1/7)(x + 10)

For -3 < x ≤ 3:
The two points in this segment are (-3, -5) and (3, 2). Calculate the slope:
m = (2 - (-5)) / (3 - (-3)) = (2 + 5) / (3 + 3) = 7 / 6

Now, use the point-slope form to find the equation for this segment:
y - (-5) = (7/6)(x - (-3))
y + 5 = (7/6)(x + 3)

For 3 < x ≤ 10:
The two points in this segment are (3, 2) and (10, 2). Calculate the slope:
m = (2 - 2) / (10 - 3) = 0 / 7 = 0

In this case, since the slope is 0, the equation is simply y = 2.

So, the piecewise linear equation that models the data is:

For -10 ≤ x ≤ -3:
y = (-1/7)(x + 10) + 8

For -3 < x ≤ 3:
y = (7/6)(x + 3) - 5

For 3 < x ≤ 10:
y = 2