## EmanAbulmagd one year ago I'm a Beginner and i have some questions please help me :) in the equation of (y-y0)=m(x-x0) the y represent the y coordinate of the tangent point to the curve and the x represents the x coordinate of the tangent point to the curve !! am i right?? and what does the (y0) and (x0) represent and what is thier importance in the equation?? thank you ^_^

1. korbi

the equation (y-y0)=m(x-x0) is the general equation of any line. To get the specific equation of a tangent line to the curve through the point P, we set x0 to the x coordinate of P and y0 to the y coordinate of P. We then use a Point Q, which is also on the curve to which we are searching a tangent line to, and set x to the x coordinate of Q and y to the y coordinate of Q. We then can solve the equation for m and thus know the slope of the tangent line. The closer Q is to P, the more precise is our result. For example: We search for a tangent line to $y=x ^{2}$ through P(1/1). Another point on the curve would be Q(2/4). So we fill those points in the equation: $(4-1) = m (2-1)$ we solve for m and get 3 as the slope of the tangent line. (it is not very precise as Q and P are not really close) To get the full equation of the tangent line, we use the general equation of a line and fill in the point P and the slope m which is 3 in this case. $(y-1)=3(x-1)$

2. EmanAbulmagd

i can never thank you enough !! ^_^ i understood it perfectly well :D but i couldn't get that paragraph " we solve for m and get 3 as the slope of the tangent line. (it is not very precise as Q and P are not really close) To get the full equation of the tangent line, we use the general equation of a line and fill in the point P and the slope m which is 3 in this case. (y−1)=3(x−1)" We get 3 as a slope....i got lost in the following lines !!

3. korbi

To get a tangent line, we start by searching for a secant. What I did is basically calculating not a tangent, but a secant line Through the points P and Q. And the smaller the distance between P and Q, the better the secant resembles the tangent we are searching for. In my example, the distance between P and Q is $\sqrt{10}$, which is quite a lot, and thus, my result, 3, is an approximation, but it could be greatly improved if we had chosen P and Q closer together. (I didn't because I wanted to keep the numbers simple). Now, I wanted to provide the equation of the tangent/secant line I was searching for. To do that, I used the general equation of any line, (y-y0)=m(x-x0), and filled in our data: We know that the line's slope m is 3, and we know that the point P(1/1) is on the line -> y0=1; x0=1; m=3. (y-1)=3(x-1)

4. creeksider

Here's a different perspective. This is an excellent question because it gets right to the heart of something that often causes confusion. How do we deal with an equation that seems to have five variables? The answer is that it really has only two variables. Three of the symbols that look like variables are actually placeholders for constants. We’re using symbols to represent those constants as a way of generalizing a solution. In other words, we want a formula that solves a whole class of problems rather than just one problem. In this case, we’re trying to solve a problem in which we’re given three numbers -- the slope of a line, the x-coordinate of a point on that line, and the y-coordinate of that point -- and our task is to find an equation for that line. For example, we might be told the slope is 2, and the line goes through a point with x-coordinate 3 and y-coordinate 5. After a little work we can come up with an equation that satisfies those conditions. Then we get another three numbers: slope = -3, x-coordinate = 2, y-coordinate = 17. Again we work with these three numbers and come up with an answer. After solving problems of this kind many times we see that it would be easier to have a template we can use to plug in these three numbers and get a solution right away. In this template we need placeholders for the three constants we’re going to be working with. We could produce a perfectly valid template in which they’re called a, b and c, but we can reduce the potential for confusion, and produce a more revealing template, if we give these placeholders names that correspond with how they’re used. We’ll use the traditional symbol for slope, which is m, and we’ll use x with a subscript to designate the constant we’re given for the x-coordinate of the point and y with a subscript for the y-coordinate. We could use any subscript here, but the choice of 0 is conventional. Using the same subscript for x and y tells us that these constants are related: they’re coordinates of the same point. And here’s what we get:$y-y_0=m(x-x_0)$So now, any time we’re given the slope of a line and the coordinates of a point on the line, all we have to do to produce an equation for that line is plug those constants into this template where the corresponding placeholders appear. No more thinking required! In calculus you’ll be dealing with many equations that use placeholders for constants. There’s no hard and fast rule that tells you which of the letters in an equation are variables and which represent constants. Sometimes this will be stated as part of the problem, but other times you have to determine this from the context of the problem. A subscript often indicates a constant, but in the equation above we also have m without a subscript as a constant. We know m is a constant because this is the point-slope formula for the equation of a line, the one we use to find the equation when we’re given the slope and the coordinates of one of the points on the line.

5. EmanAbulmagd

@creeksider in the previous answer korbi said that in the tangent line x0 represents the x coordinate of P and y0 represent the Y coordinate of P ! from the prespective of the placeholders if the equation goes like this P(1,1) and Q(2,4) and the slope is 3 which of them are the placeholders ?? i understood that the place holder is a constant for a certain graph we're dealing with !! but not constant for other graphs!! am i right?

6. EmanAbulmagd

i mean constant for a certain tangent !! as P is constant in the graph and the point Q's change represents the movement of secant!!

7. creeksider

The confusion here comes from mixing up two different concepts. The equation you presented was$y-y_0=m(x-x_0)$This is called the point-slope form of the equation for a line, and it is used when you know the coordinates of a point and the slope of a line. The first response above talks about the process by which we find the equation of a line that's tangent to a curve at a particular point. In this situation we have the coordinates of a point and a function that produces the curve but we don't know the slope, so we aren't ready to use the point-slope form of the equation. We have to find out what the slope is before we can use the point-slope form to produce the equation. As we begin the study of calculus, we approach this problem by calculating the slope of secant lines -- in other words, lines that connect two different points on the curve we're studying. One of the points will be the one where we want to find the tangent, and the other will be some nearby point. In this calculation we don't know the slope but instead we know the coordinates of two different points on the line. We have the coordinates of one point because they are given, and the coordinates of the other point come from choosing a nearby x and then applying the function to that x-value to determine the y-value. The equation could be written this way:$m=\frac{ y_1-y_0 }{ x_1-x_0 }$Now we have four constants (the coordinates of the two points), and in this equation m functions as a variable, not a constant. As I said in my earlier response, the context in which we use a letter determines whether we interpret it as a constant or a variable. You'll see that in calculus we change the way we write the equation for calculating slope. In the denominator we replace x_1 - x_0 with Delta x, which means the same thing. In the numerator, the values for the y-coordinates are shown as the results of applying the function we're studying to the x values. So the slope of a secant line looks like this:$\frac{ f(x_0+\Delta x)-f(x_0) }{ \Delta x }$The main point here, in response to your original question, is that x0 and y0 are placeholders for constants. They tell you where to plug in the values for the x- and y-coordinates of a particular point when those are given.

8. EmanAbulmagd

@creeksider I was really puzzled thank you very much i got it ^_^ :D