Question

interpolation and extrapolation
besides looking at the data thats visible on a graph, you can make whats not explicitly available through your data
interpolation is a process in which you estimate a value that lies between two known values on a
graph. you start with two known points as inputs, such as ((x_0,y_0)) and ((x_1,y_1)). then, for a point between
those two known points, you can determine the value of one variable (y) when given the value
of the other variable (x).
the most common interpolation approach is to assume that all points lie on a straight line. using
this equation,
y - y_0 = left \frac{y_1 - y_0}{x_1 - x_0}
ight \times (x - x_0)
you can input any value of x between the known points and get an estimated value for y
example
to stretch a spring, we need to apply force.
the table and the graph show data for the amount
of force (f) required to stretch, or elongate (x), a
particular spring.
from the table, wed guess that a force of
2.5 newtons would extend the spring by
1.25 centimeters.
lets interpolate between ( f = 2.0 ) newtons and ( f = 3.0 ) newtons to get this value. substituting
the y and force (f) for x in the interpolation formula, we get these calculations:
x - x_0 = left \frac{x_1 - x_0}{f_1 - f_0}
ight \times (f - f_0)
x - 1.0 = left \frac{1.5 - 1.0}{3.0 - 2.0}
ight \times (2.5 - 2.0)
x - 1.0 = \frac{0.5}{1} \times 0.5
( x = 1.25 ) centimeters

Explanation:

Step1: Calculate the numerator and denominator in the fraction

The fraction is \(\frac{(1.5 - 1.0)}{(3.0 - 2.0)}\).
\(1.5−1.0 = 0.5\) and \(3.0−2.0=1.0\), so \(\frac{0.5}{1.0}=0.5\)

Step2: Calculate the value of \((F - F_0)\)

\(F = 2.5\) and \(F_0 = 2.0\), so \(F - F_0=2.5 - 2.0 = 0.5\)

Step3: Calculate \(x-1.0\)

\(x - 1.0=0.5\times0.5 = 0.25\)

Step4: Calculate \(x\)

\(x=1.0 + 0.25=1.25\)

Answer:

\(1.25\)