Find the equations for the piecewise function ft shown in Fi

Find the equations for the piecewise function f(t) shown in Figure 6. The function f(t) in the interval (-1, 5) is composed of four segments: two first-order polynomials and two second-order polynomials.

Solution

Equation for first segment in the interval (-1,0) is :-

It is a straight line. Points are (-1,1) and (0,3)

(y-y₁) = m (x-x₁) where m = (y₂-y₁)/(x₂-x₁)

(y-1) = (2/1) (x-(-1))

y-1 = 2(x+1)

2x - y + 3 = 0

Replace y with f(x)

So, equation is f(x) = 2x + 3

Equation for second segment in the interval (0,2) is :-

It is a parabola.So, (h,k) = (1,2) and a = 1

Equation of parabola is y = a(x-h)² + k

y = 1 (x-1)² + 2

Replace y with f(x)

So,equation is f(x)= x² -2x + 3

Equation for third segment in the interval (2,4) is :-

It is also a parabola. (h,k) = (3,2) and a = 3

Equation of parabola is y =a(x-h)² + k

f(x) = 3(x-3)² + 2

f(x) = 3x² - 18x + 29

Equation for fourth segment in the interval (4,5) is :-

It is a straight line. Points are (4,3) and (5,1)

(f(x)-3) = ((-2)/1) (x-4)

f(x) = -2x + 8 + 3

So,equation is f(x) = (-2x + 11)