A continuous function is a function that has no gaps. You could draw it without picking up your pencil. The pertinent points are at the boundaries with x = 1 and x = 2.
First, x = 1
f(x) = 4x
f(1) = 4
f(x) = cx2 + d also has to equal 4 at x = 1
4 = c(1)2+d
4 = c + d
Now let's look at x = 2
f(x) = 7x
f(2) = 7×2
f(2) = 14
f(x) = cx2 + d also has to equal 14 at x = 2
14 = c(2)2 + d
14 = 4c + d
Now we have a system of equations with two variables.
4 = c + d
14 = 4c + d
There are many methods to solve this. I am going to use substitution.
4 = c + d Subtract c from both sides
4 - c = d
d= 4 - c
14 = 4c + d Substitute for d
14 = 4c + 4 - c Combine like terms
14 = 3c + 4 Subtract 4 from both sides
10 = 3c Divide both sides by 3
10/3 = c
Go back to the original equations and substitute 10/3 for c
4 = c + d
4 = 10/3 + d Change 4 to thirds
12/3 = 10/3 + d Subtract 10/3 from both sides
2/3 = d
f(x) = 10/3 x2 + 2/3
Garrett S.
10/07/18