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Slope of a line secant to a curve

What is the slope of a line between two points on a curve?  This is called a "secant" line. Created by Sal Khan.

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  • male robot hal style avatar for user Scott Fitzgerald
    I'm having difficulty understanding the concept of a secant line as it pertains to a sine graph. Specifically, there is a problem in the "Slope of secant lines" exercise, where there are four questions. In each you are asked to evaluate the rates of change between secant lines for four different points.

    The second question asks whether [Sin 5/2 Pi - Sin 1/2 Pi / 5/2 Pi - 1/2 Pi] is greater than, less than or equal to [Sin 2/3 Pi - Sin 1/3 Pi / 2/3 Pi - 1/3 Pi]. Evaluating the first part, I see that 5/2 Pi is the same as 1/2 Pi, and therefore the change in Y is zero. However, when evaluating the second part, it seems to me that Delta Y/Delta X is Sin 1/3 Pi / 1/3 Pi.

    But the lesson tells me that the secant line is horizontal, which I do not understand. Can anyone help me with understanding secant lines on a sine graph? I did all of the trig lessons and most of the videos already, but if anyone can clarify this concept or point me in the direction of some valuable review materials, that would be legit. ( :
    (17 votes)
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    • leaf green style avatar for user ArDeeJ
      You're correct that sin(5π/2) - sin(π/2) = 0, so the slope of the first line is 0 making it horizontal.

      Let's look at the second slope. In the numerator it has sin(2π/3) - sin(π/3).
      Can we simplify that?

      Remember that sin x = sin(π - x), so
      sin(2π/3) = sin(π - 2π/3) = sin(π/3).

      The numerator then becomes
      sin(2π/3) - sin(π/3) = sin(π/3) - sin(π/3) = 0.

      So, both slopes are horizontal and thus equal.

      Maybe the mistake you made was that you thought that
      sin(2π/3) - sin(π-3) = sin(2π/3 - π/3)?
      The two aren't equal.
      (22 votes)
  • leafers seedling style avatar for user Sylvia
    Does a secant line always only touch 2 points on the graph? (is this not doable for some periodic functions where it would hit it 3 times + for example?)
    (8 votes)
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    • piceratops ultimate style avatar for user Just Keith
      As the term is typically used in calculus, a secant line intersects the curve in two places locally -- it may or may not intersect the curve somewhere else. So the requirement of just two intersections applies just to the small region of interest and is not a strict requirement for regions you are not concerned with at the moment.

      Note: in some fields of geometry the requirement of exactly two points of intersection is much more strict than what we usually have in calculus.
      (17 votes)
  • duskpin ultimate style avatar for user Rusab Baig
    what is the difference between a chord and a secant?
    (2 votes)
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  • blobby green style avatar for user Horacio Prado
    What are the main differences between a tangent line and a secant line in a curve?
    (2 votes)
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  • aqualine seed style avatar for user David Richards
    Why does he use x not and sub 1 in the first graph but sub 1 and sub 2 in the second graph? Is there any special situation that those are reserved for?
    (4 votes)
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  • purple pi pink style avatar for user Barsha Chaudhary
    Can't we calculate slope in the same way as we do for lines?We have coordinates in the curve.
    (3 votes)
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  • purple pi purple style avatar for user Jakub
    At Sal mentions interval. Can someone explain to me what this 'interval' means? I keep on hearing this term more and more often.
    (0 votes)
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  • duskpin ultimate style avatar for user Nonu
    Can't we call that secant line is a chord? Or is it both? Or one in specific?
    (1 vote)
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  • aqualine ultimate style avatar for user Sarvesh Parakh
    In the end, he ends up with finding the slope of a line with points (X0, Y0), (X1, Y1). This way, when I know two points on the line, I can find out the equation of that line. Now, that line, suppose line m, was secant to the curve, suppose L. So Line m is secant to the curve L. Now, let's imagine a line n, which is parallel to line m and tangent to the curve L. So i know the slope of the line m is equal to the slope of the line n, because they are parallel. Hence, can I find out the point at which the tangent n intersects the curve L using just algebra? I think I can, using some concepts in co-ordinate geometry. Suppose that point is (X3, Y3). So without actually using Differential calculus, we have the value of the instantaneous point. So if this was the example of a car running on a road and a graph or it's speed on y and time on x axis, then i guess that at X3, dy/dx = Y3....i.e. instantaneous velocity at X3 = Y3. So we solved it without using calculus. Someone please tell me if i'm correct or wrong!
    (2 votes)
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    • spunky sam blue style avatar for user Kali Reagan
      The problem is that the slope of the tangent line, while it may be very close, is not exactly the same as the secant line. In the interval between the two point where the secant line and curve are the same, there are an infinite number of tangent lines each with a slightly different slope. You can use algebra to estimate a tangent line, but to actually find the tangent line, you need to find the limit of the slope of the secant lines which is calculus.
      (2 votes)
  • leaf green style avatar for user sst
    A secant gives the average rate of change of a function. However, this is not the accurate value of the average change. How to find out the average rate of change of an arbitrary function?
    (1 vote)
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    • leaf green style avatar for user jude4A
      You can use differential calculus to determine the rate of change of functions at given points. So for instance, if I want to know what is the slope of a line tangent to a part of sinx, I need to be able to calculate its derivative.
      (4 votes)

Video transcript

So let's review the idea of slope, which you might remember from your algebra classes. The slope is just the rate of change of a line. Or the rate of change of y, with respect to x, as we go along a line. And you could also view it as a measure of the inclination of a line. So the more incline the line is, the more positive of a slope it would have. So this right over here, this has a positive slope. It's increasing as x increases. And if this had an even higher inclination like this, if it increased even more as x increased, then it would even have a higher slope. So this right over here is some line. So that's some line. And just as a reminder, we can figure out the slope between two points. Two points define a line. And between those two points, we can find the rate of change of y, with respect to x. So let's put two points on here. So let's say that this point right over here, this x value, is x sub-- well, this is pronounced x naught, or x sub 0 is just x naught-- and when x is x0 for this line, y is y0. So this is a point x0, comma y0. And let's say we have another point all the way over here. And let's say that this x value, this x value right over here is x sub 1. And the y value over here is y sub 1. So this is the point x sub 1, y sub 1. So just as a review, the slope of this line, and a line by definition, has a constant slope between any two points that you pick. The slope of this line, which is often denoted by the letter m, is your rate of change of y with respect to x. Or another way of thinking about it, for a given change in x, how much are you changing y? Or a change in y divided by a change in x. Just as a reminder, this triangle, that's the Greek letter delta. It's shorthand for change in whatever. So change in y over change in x. So let's think about what this is going to be for this example right over here. Well, let's think about change in x, first. So we are moving from x0 to x1. So our change in x. So this is our change in x, right over here. We're starting at x0 and going to x1. That is our change in x. I'll put that in that pink color. That is our change in x. And what is it equal to? Well if we're ending here and we started here, let's just do ending point minus starting point. So it is x1 minus x0. And that way, doing it this way, I would have made sure that I have a positive value. I'm just assuming that x1 is larger than x0. And what is my change in y? Well, once again, ending point, ending y value minus starting y value. y1 minus y0. Now, you might be saying hey, could I have done y0 minus y1 over x0 zero minus x1? Absolutely. You could have done that. Then you would have just gotten the negative of each of these values in the numerator and denominator, but they would have canceled out. The important thing is that you're consistent. If you're subtracting you're starting value from your ending value in the numerator, you have to subtract your starting value from your ending value in the denominator as well. So this right here you probably remember from algebra class. The definition of slope is the rate of change of y with respect to x. Or it's the rate of change of our vertical axis, I should say, with respect to our horizontal axis. Or change in y, or change in our vertical axis over change in a horizontal axis. Now I'm going to introduce a little bit of a conundrum. So let me draw another axis right over here. Scroll over a little bit just so we have some space to work with. So that was for a line. And a line, by definition, has a constant slope. If you calculate this between any two points on the line, it's going to be constant for that line. But what happens when we start dealing with curves? ' When we start dealing with non-line or non-linear curves. So let's imagine a curve that looks something like this. So what is the rate of change of y with respect to x of this curve? Well, let's look at it at different points. And we could at least try to approximate what it might be in any moment. So let's say that this is one point on a curve. Let's call that x1, and then this is y1. And let's say that this is another point on a curve right over here, x2. And let's call this y2. So this is a point x1, y1, this is a point x2, y2. So we don't have the tools yet. And this is what's exciting about calculus, we will soon have the tools to figure out, what is the rate of change of y with respect to x at exactly this point? But we don't have that tool yet. But using just the tools from algebra, we could at least start to think about, what is the average rate of change over the interval from x1 to x2? Well, what's the average rate of change? Well, that's just how much did my y change-- so that's my change in y-- for this change in x. And so we would calculate it the same way. y2 minus y1 over x2 minus x1. So our change in y over this interval is equal to y2 minus y1, and our change in x is going to be equal to x2 minus x1. So just like that we were able to figure out the rate of change between these two points. Or another way of thinking about it is, this is the average rate of change for the curve between x equals x1, and x is equal to x2. This is the average rate of change of y with respect to x over this interval. But what if we have we also figured out here? Well, we figured out the slope of the line that connects these two points. And what we call a line that intersects a curve in exactly two places? Well, we figured out, we call that a secant line. So this right over here is a secant line. So the big idea here is we're extending the idea of slope. We said, OK, we already knew how to find the slope of a line. A curve, we don't have the tools yet, but calculus is about to give it to us. But let's just use our algebraic tools. We can at least figure out the average rate of change of a curve, or a function, over an interval. That is the same exact thing as the slope of a secant line. Now just as a little bit of foreshadowing, where is this all going? How will we eventually get the tools, so that we can figure out the instantaneous rate of change, not just the average? Well just imagine what happens if this point right over here got closer and closer to this point. Then the secant line is going to better and better and better approximate the instantaneous rate of change right over here. Or you could even think of it as the slope of the tangent line.