Question

Find the linear function formula f(x) = mx + b for the function described.A) f(2) =6 and f(4) =0B) f(3)=4 and f(6) = -3

Answer 1

Given:

A)

`\begin{gathered} f(2)=6 \\ f(4)=0 \end{gathered}`

It can be written as,

`\begin{gathered} (x_1,y_1)=(2,6),(x_2,y_2)=(4,0) \\ \text{Use the two point form to find the linear equation,} \\ (y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1) \\ (y-6)/(0-6)=(x-2)/(4-2) \\ (y-6)/(-6)=(x-2)/(2) \\ 2(y-6)=-6(x-2) \\ 2y-12=-6x+12 \\ 2y=-6x+12+12 \\ 2y=-6x+24 \\ y=-(6)/(2)x+(24)/(2) \\ y=-3x+12 \end{gathered}`

Answer:

`f(x)=-3x+12`

B)

`\begin{gathered} (x_1,y_1)=(3,4),(x_2,y_2)=(6,-3) \\ To\text{ find the linear equation f(x)=mx+b} \\ \text{Here, m is slope and b is y intercept.} \\ \text{First find slope.} \\ m=(y_2-y_1)/(x_2-x_1)=(-3-4)/(6-3)=(-7)/(3) \\ \text{Now put (x,y) = ( 3,4) in the equation y=f(x)=mx+b} \\ y=mx+b \\ 4=-(7)/(3)(3)+b \\ 4=-7+b \\ b=4+7 \\ b=11 \\ So,\text{ the linear equation f(x)=mx+b is,} \\ f(x)=-(7)/(3)x+11 \end{gathered}`

Answer:

`f(x)=-(7)/(3)x+11`