Question

Suppose that `f` is a linear function such that `f(3) = 6` and `f(-2) = 1`. What is `f(8)`?

Answer 1

`f(8)=11`

Explanation:

Since it's a linear function, it must be of the form

`ax+b=0" " " "(1)`

So

`f(3) = 3a + b = 6`

`f(-2) = -2a + b = 1`

Solving for `a` and `b` gives `1` and `3`, respectively.

Therefore, substituting the values of `a`, `b`, and `x=8` in equation `(1)` gives

`f(8) = 1 * 8 + 3 =11`

Answer 2

`f(8)=11`

A lot more explanation is involved than doing the actual maths

Explanation:

Linear basically means 'in line'. This is implying a strait line graph situation

You read left to right on the x-axis so the first value is the least `x`
using:

`f(-2)=y_1=1`
`f(3)=y_2=6`
`f(8)=y_3 ="Unknown"`

Set point 1 as `P_1->(x_1,y_1)=(-2,1)`
Set point 2 as `P_2->(x_2,y_2)=(3,6)`
Set point 2 as `P_3->(x_3,y_3)=(8,y_3)`

The gradient (slope) of part will be the same gradient of the whole.

Gradient (slope) is the amount of up or down for a given amount of along, reading left to right.

Thus the gradient gives us: `P_1->P_2`

`("change in "y)/("change in "x) ->(y_2-y_1)/(x_2-x_1) =(6-1)/[3-(-2)]=5/5`

Thus we have `P_1->P_3` ( same ratio )

`("change in "y)/("change in "x) ->(y_3-y_1)/(x_3-x_1) =(y_3-1)/[8-(-2)]=5/5`

`color(white)("dddddddd")-> color(white)("ddd")(y_3-y_1)/(x_3-x_1) =color(white)("d")(y_3-1)/10color(white)("d")=1`

Multiply both sides by 10

`color(white)("dddddddd")->color(white)("dddddddddddddd")y_3-1color(white)("d")=10`

Add 1 to both sides

`color(white)("dddddddd")->color(white)("ddddddddddddddddd")y_3color(white)("d")=11`