# Linear interpolation formula

In the following lecture:

http://www.ece.mcmaster.ca/~xwu/interp_1.pdf

the model (formula) for solving the linear interpolation problem (1D) given at p.5 is:

$$f(x)= a_1x_1 + a_0x_0$$

solve for $$a1,a0$$

the system:

$$f(0)=a_i*0 + a_0*1$$

and

$$f(1)=a_i*1 + a_0*1$$

I don't understand where this formula comes from. Even though deriving in all possible ways I could think of, I cannot get the equation of a line $$y=mx+b$$ ( also written as: $$f(x)= mx+b$$, where m is the slope and b is the y-intercept) to look something like $$f(x)= a_1x_1 + a_0x_0$$. Any idea/proposals/discussions etc.?

To my eye, the equations in the slides look like $$f(x) = a_1 x^1 + a_0 x^0 = a_1 x + a_0$$. So it's an exponent on the $$x$$, not an index. As long as you are interpolating linear, the only exponents that you encouter are 1 (linear) and 0 (constant). When you go to quadratic, you'll see $$x^2$$ terms appearing. Now, $$a_1$$ is the coefficient of the linear term (your slope $$m$$) and $$a_0$$ is the coefficient of the constant term (the y-intercept $$b$$).
This should explain where the system of equations for $$a_1$$ and $$a_0$$ comes from.