This exploration will look at parametric equation. "A parametric equation is an equation that introduces a third variable, t, which is called a parameter. By writing both x and y as functions of t, you obtain a parametric equation." -- This paragraph was taken from Calculus by Larson, et.al., 5th edition.
Now let us consider the parametric equation
We will start off by letting a = b = 0 and look at the two different values for k.
Graphing these equations for t > 0 we see that the equations start at the origin and are linear equations with k equal to the slopes of the lines.
If we solve for t in terms of x in the original equation and plug that value into the y function we get
We can see that this parametric equation is actually a linear function with slope k and y-intercept (b - ka). Here is an example of another parametric equation: