Let’s say you have a function, y = f(x), where y is a function of x. It doesn’t matter what the specific relationship is. It could be y = x^2, for example, a simple and familiar parabola passing through the origin. It could be y = x^2 + 1, a parabola with an identical shape and a vertex one unit above the origin. It could be a more complex function, such as y = x^3. Regardless of what the function is, a straight line passing through any two points on the curve is a secant line.
Notice that the secant line changes as you pick a second point closer to the first point. You can always pick a point on the curve closer than you did before and get a new secant line. As your second point gets closer and closer to your first point, the secant line between the two approaches the tangent to the curve at the first point.
Take the x and y values for any two points you know to be on the curve. Points are given as (x value, y value), so the point (0, 1) means the point on the Cartesian plane where x = 0 and y = 1. The curve y = x^2 + 1 contains the point (0, 1). It also contains the point (2, 5). You can confirm this by plugging each pair of values for x and y into the equation and ensuring that the equation balances both times: 1 = 0 + 1, 5 = 2^2 + 1. Both (0, 1) and (2, 5) are points of the curve y = x^2 +1. A straight line between them is a secant and both (0, 1) and (2, 5) will also be part of this straight line.
Determine the equation for the straight line passing through both these points by choosing values that satisfy the equation y = mx + b -- the general equation for any straight line -- for both points. You already know that y = 1 when x is 0. That means 1 = 0 + b. So b must be equal to 1.
Substitute the values for x and y at the second point into the equation y = mx + b. You know y = 5 when x = 2 and you know b = 1. That gives you 5 = m(2) + 1. So m must equal 2. Now you know both m and b. The secant line between (0, 1) and (2, 5) is y = 2x + 1
Pick a different pair of points on your curve and you can determine a new secant line. On the same curve, y = x^2 + 1, you could take the point (0, 1) as you did before, but this time select (1, 2) as the second point. Put (1, 2) into the equation for the curve and you get 2 = 1^2 + 1, which is obviously correct, so you know (1, 2) is also on the same curve. The secant line between these two points is y = mx + b: Putting 0 and 1 in for x and y, you’ll get: 1 = m(0) + b, so b is still equal to one. Plugging in the value for the new point, (1, 2) gives you 2 = mx + 1, which balances if m is equal to 1. The equation for the secant line between (0, 1) and (1, 2) is y = x + 1.
- Notice that the secant line changes as you pick a second point closer to the first point. You can always pick a point on the curve closer than you did before and get a new secant line. As your second point gets closer and closer to your first point, the secant line between the two approaches the tangent to the curve at the first point.
About the Author
Andrew Breslin has been writing professionally since 1994. His articles and op-ed pieces have appeared in the "South Florida Sun Sentinel," "St Paul Pioneer Press," "Detroit Free Press," "Charlotte Observer," “Good Medicine,” and others. He studied molecular biology at Westchester University and frequently writes about science and mathematics.