Parametric Equations

by Michael Walliser

Consider the following set of parametric equations:

**x = a + t**

**y = b + kt**

Let's see what happens when we graph these equations with varying values of **a**, **b**, and **k**. We'll start by holding **a** and **b** constant, let's say at **a = 1** and **b = 2**, and using **k** values of -2, -1, -1/2, 0, 1/2, 1, and 2.

k = -2 ______ k = -1 ______ k = -1/2 ______ k = 0 ______ k = 1/2 ______ k = 1 ________ k = 2

We see that **k** simply represents the slope of the curve. This makes sense when we consider that, for any change in **t**, **y** changes at **k** times the rate that **x** changes. Be aware that if we multiply **t** by a coefficient in the **x** equation, then the slope will be the ratio of **k** to that coefficient. Notice also that all the curves pass through the point (1,2). This point is common to all the curves when **t = 0**, so unsurprisingly, it can be represented more generally as **(a,b)**. Let's check this hypothesis by varying the values of **a** and **b**.

The following graphs uses the same **k** values, but we now set **a = -3** and **b = 1** in the graph on the left, and **a = 2** and **b = -1.5** in the graph on the right.

k = -2 ______ k = -1 ______ k = -1/2 ______ k = 0 ______ k = 1/2 ______ k = 1 ________ k = 2

Indeed, the slopes stay the same, and the curves are concurrent at **(a,b)** in each graph.

If we want to model these graphs using **y** as a function of **x**, we can simply solve the **x** equation for **t**, then plug the solution into the **y** equation. For instance, we can rearrange **x = a + t** to say that** t = x - a**. Thus, we can rewrite **y = b + kt** as **y = b + k(x - a)**, or **y = kx + (b - ka)**, where **k** is the slope and **b - ka** is the y-intercept. If we observe our graphs above, we see that these slope-intercept equations hold for all values of **k**, **a**, and **b**.