Polynomial Cubic Equations

[LisaJean]

"A function passes through the points (2,4) and (6,10). Determine the equation of a cubic that passes through these points and graph it using the restricted domain."
I have attempted this problem as an inequality but because I only have two points I can seem to get very far. I know you can set it up using f'(x) but I haven't taken calculus yet so I can't solve it that way.
(2,4) (6,10)
I've set it up this way but I'm not sure if this is correct
4=8a+4b+2c+d
10=216a+36b+6c+d
this is where I get stuck!

 

[Ishuda]

"A function passes through the points (2,4) and (6,10). Determine the equation of a cubic that passes through these points and graph it using the restricted domain."
I have attempted this problem as an inequality but because I only have two points I can seem to get very far. I know you can set it up using f'(x) but I haven't taken calculus yet so I can't solve it that way.
(2,4) (6,10)
I've set it up this way but I'm not sure if this is correct
4=8a+4b+2c+d
10=216a+36b+6c+d
this is where I get stuck!

First, since you are given two points you must go through, set up the linear equation (a straight line) passing through the two points. Call that function g(x). That is g(2)=4 and g(6)=10. Also let f(x) be the function we want. If we write
f(x) = g(x) +(x-2)(x-6)h(x)
where h(x) is any function we want to choose then f(2)=g(2)=4 and f(6)=g(6)=10. So we have our function f except for one last requirement.

What would h(x) have to be if f(x) were a cubic?

EDIT: One reason why you may be having problems the way you approached it is that you have 4 coefficients (the a, b, c, and d) but only two equations. In general this means you can pick any two (4 - 2) coefficients and set them to what you would like. In the case I showed of how to solve the problem, I picked a different particular form of a cubic so that I could then arbitrarily choose an equivalent a and b [as long as a was not zero].

To do that the way you were doing the problem, rewrite your equations
2c+d = 4 -8a-4b
6c+d = 10-216a-36b
Now solve your equations for c and d in terms of a and b. Then choose your a and b. Or you could just choose a=1 and b=0 and then solve.