Secant Line: Definition & Formula
In this lesson, we’ll learn what a secant line is and how to find its equation. We’ll also look at some examples of secant lines in the world around us.
What Is a Secant Line?
When we draw a line on the graph of a curve, three things can happen:
The line does not intersect the curve.
The line intersects the curve at exactly one point.
The line intersects the curve at two or more points.
Number three describes a secant line. In mathematics, a secant line is a line that intersects a curve in two or more places. To illustrate this, observe the graph of y = x^2 with a secant line displayed below, where x represents the graph’s horizontal line and y stands for the vertical line.
secant line
We can observe secant lines in the world around us. Anywhere we see a curve with a line intersecting it at two or more points, we have a secant line. For instance, look at these images below of a bridge and a bicycle wheel:
Secant line examples
In the bridge picture, we see that the arch can be seen as a curve shown in pink, and the road is a secant line shown in green. We see that the road intersects the arch in two different places. Similarly, in the wheel picture, the wheel is a circular curve shown in pink, and the spokes are secant lines, one highlighted in green. The spokes intersect the wheel in two different places.
Tthe Equation of a Secant Line
As we’ve just learned, a secant line intersects a curve at two or more points. In mathematics, when we are given two points, call them (x1, y1) and (x2, y2), we can find the slope of the line through these two points using the formula (y2 – y1) / (x2 – x1). As a quick reminder, the slope of a line is the rate of change of y with respect to x, hence the formula:
(Change in y) / (Change in x) = (y2 – y1) / (x2 – x1)
Once we have found the slope of the line through these two points, we can find the equation of the line through these two points by plugging one of our points (x1, y1) and our slope m into the equation:
y – y1 = m(x – x1)
This equation is called the point slope form of a line.
Therefore, if we can find two points on our secant line, we can find the equation of that line. To do this, we follow these steps:
Find two points on the secant line
Find the slope of the line between the two points
Plug one of your points and your slope into the point slope form of your line to obtain an equation of the line
Let’s look at an example to illustrate these steps:
Secant line
This is a graph of y = -x^2 + 4 with a secant line that passes through the points on the curve where x = -1 and x = 2. We want to find the equation of the secant line, so we follow our steps:
1.) Find two points on the secant line: We have our x-values of our two points. To find their corresponding y-values, we recognize that these two points are on the curve:
y = -x^2 + 4
so we can plug our x-value into this equation to get our corresponding y-value. When x = -1, we have:
y = -(-1)^2 + 4 = 3
Thus, one of our points is (-1,3). When x = 2, we have:
y = -(2)^2 + 4 = 0
so our other point is (2,0).
2.) Next, find the slope of the line between the two points: We now have our two points on our secant line, (-1,3) and (2,0). To find the slope between them, we plug them into our slope formula, which – remember – is:
(y2 – y1) / (x2 – x1)
and we get:
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