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

2 Answers
Jan 23, 2018

#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#

Jan 23, 2018

#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

Tony B

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#