# Finding the Equation of A Graph using Tangents

• healey.cj
In summary, the conversation discusses a question on a math assignment that involves fitting a quadratic equation to three points on a cartesian plane. The scenario involves a cup of hot coffee cooling over a 50 minute period, with a table of values given. The conversation includes discussions on using derivatives and finding the equation of the quadratic function, as well as clarifying the instructions given in the problem. Ultimately, the recommended approach is to use the given 3 points to solve for the coefficients of the quadratic equation.
healey.cj
Hey everyone,

We've been covering tangents and derivatives etc in class recently but there is a question on the assignment that we've been given that i don't know how to do.

The question is:
"A quadratic equation can be fitted to any three points on a cartesian plane. The model for such an equation in y=ax + bx+ c. By substitution three separate points (The first, middle and last data points) derive the equation of the Quadratic function that models the data."

The situation is a cup of hot coffee is left to cool over a 50 minute period and we have been given this table of values:

mins : Degrees Celcius
0 :83
5 :76.5
8 :70.5
11:65
15:61
18:57.5
24:52.5
35: 51
30:47.5
34:45
38:43
42:41
45:39.5
50:38

From there i have plotted the time vs. temp graph and attached 3 tangents to points time (t) = 0, t = 25 & t = 50.

at t=0 the Rate of Change or gradient was -1.85Degrees/min
at t=25, the ROC or Gradient was -0.92degrees/min
at t=50, the ROC or Gradient was -0.13degrees/min

I'm not asking for you to do this for me, I am just asking for the path i need to take or an idea of what i need to do...

Thanks everyone,
Chris

yeah, the derivatives you have stated are correct...

What the...where did the other post just go?

I can only think you should use the derivative of the general formula with 2 gradients to solve for a and b, then peg c not with a gradient but a coordinate.

Wait, they specifically say "by substituting 3 separate points, derive the formula", so I don't think you are allowed to use the derivative. So just use 3 points to get 3 equations and hopefully they are easy to solve.

How about doing what you were told to do in the problem? Is that to complicated?

You are told to use a quadratic: y= ax2+ bx+ c. You are also told to use a specific 3 points: "(The first, middle and last data points)". The only problem is that there is NO "middle" point since you are given 14 points. I would be inclined to use halfway between the 7th and 8th points. Put those 3 values in for x and y and solve for the coefficients. I can see no reason to worry about tangents.

## 1. What is the purpose of finding the equation of a graph using tangents?

The purpose of finding the equation of a graph using tangents is to determine the equation of a curve at a specific point. This is useful in many applications, such as physics, engineering, and economics, where the slope of a curve at a point represents an important concept or value.

## 2. How do you find the equation of a graph using tangents?

To find the equation of a graph using tangents, you first need to find the slope of the tangent line at the point of interest. This can be done by taking the derivative of the original equation. Then, you can use the point-slope form of a line to write the equation, substituting in the slope and the coordinates of the point.

## 3. What information do you need to find the equation of a graph using tangents?

To find the equation of a graph using tangents, you need the original equation of the curve, the coordinates of the point where you want to find the equation, and the slope of the tangent line at that point.

## 4. Can you find the equation of any type of graph using tangents?

Yes, the method of finding the equation of a graph using tangents can be applied to any type of curve, as long as the curve is differentiable at the point of interest.

## 5. How is finding the equation of a graph using tangents useful in real life?

Finding the equation of a graph using tangents can be useful in many real-life situations. For example, in physics, the slope of a position-time graph represents the velocity of an object at a given time. In economics, the slope of a demand or supply curve represents the rate at which the quantity changes with respect to price. In engineering, the slope of a stress-strain curve represents the stiffness of a material. Knowing the equation of a graph at a specific point can provide valuable information for analysis and decision-making.

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