Question

determine the linear function with f(-2)=4 and f(4)=1

Answer 1

let y=ax+b be the linear function then from the given data we have
4=-2a+b and 1=4a+b
just solve to get a & b

Answer 2

if \(f\) is a linear function \(f(x)=mx+b\) then it has constant slope between two points i.e.$$m=\frac{f(4)-f(-2)}{4-(-2)}=\frac{1-4}{4+2}=\frac{-3}6=-\frac12$$

Answer 3

to find the \(y\)-intercept note it's just \(f(x)-mx\) e.g. \(f(4)-4m=1-(-1/2)\cdot4=1+2=3\)

Answer 4

so our function is \(f(x)=-\dfrac12x+3\)

Answer 5

where did 1-4 come from?

Answer 6

\(f(4)=1\) and \(f(-2)=4\) so \(f(4)-f(-2)=1-4=-3\)

Answer 7

also the intercept equation is from \(f(x)=mx+b\) -- if you subtract \(mx\) from both sides you get \(b=f(x)-mx\)

Answer 8

Solve f(-2)= 4 and f(4) =1 for a and b
you will get
\[
-2 a + b = 4\\
4 a + b = 1\\

\text{ subtract them}\\
-6a =3 \\
a= -\frac 1 2\\
\text {Multiply the first equation by 2 and add them}\\
3b = 9\\
b= 3\\
y= -\frac 12 x + 3
\]