Investigation, Proof

 

Given equations

x = a + t

y = b + kt

 

Let's solve for t in the first equations

t = x - a

y = b + kt

 

Now lets plug in for t into the second equation

y = b + k * (x - a)

 

Expand this equation out...

y = b + kx -ka

 

Rearranging the equation a bit gives the form

y = kx + b - ka

 

We now have a form of a equation that is almost in y = mx + b form

y = kx + (b-ka)

 

Therefore we can see that:

k = slope

b-ka = y-intercept

 

Therefore we can go back to everyone of the equations we just experimented with and figure out and check to see if we plug in the numbers we get the coordinates that should be plotted on the graph.

One important not is that not all of the graphs were graphed with y-intercepts. This is due to the t values from -2 to 2 that were used. Making these numbers larger would allow us to see the y-intercepts of all the graphs.

 

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