Algebra-parabola FINAL
- Author:
- Alec Contreras
The parabola
We will discuss the function f(x)=2x^2+4x-3 which is defined as a parabola because it has a power of two and the inputs yield an output whose distances are equidistant from each other and which has a minimum value at this particular origin.
When taking the derivative of f(x) we're left with the equation f'(x)=g(x)= 4x+4. Which is not a quadratic and is a line that is continuous.
The releationship between g(x) and f(x) could be noticed if we were to first think of the parabola function f(x) as a the movement of a car whose position(y) changes over time(x) for a domain of infinite values greater than y= -5. In other words, first notice that the path of the graph represents an object that is hypothetically moving at a constant rate. Since the object is a car, suppose that when it approaches a very sharp and narrow turn it has to make a change to its speed by slowing down how fast it's going. In accordance to the parabola it slows down so much that it reaches a minimum at point (-1, -5) and after that it starts on going in a positive direction. This allows you only to assume that there had to be some sort of change in the path of the car but in order to know that there was a clear change, that's were f'(x) comes in.
In point slope form fr the values of (0,-3) the graph tangent to the curve is depicted in orange.
We stated that f'(x)=g(x) and it yielded a linear instead of a quadratic graph. This graph known as the derivative of the original allows us to know extactly when the object from the first parabola changed direction. By looking at the function of g(x)=4x+4 from the left we see that before the point (-1,0) all values are negative and after it the function yields positive values by being on the positive side of the y-plane. Now when we consider our example about our "car" reaching a minimum and then going fast again we can think of this interms of it's velocity while considering the rate of distance over time, or in other words we can think of how our car is changing it's miles per hour to make the sharp turn and ultimatley alter it's direction. Therefore the graph of f'(x) makes it simpler to describe an exact change that an object is undergoing (given f(x)) and we can use f'(x) to look at f(x) with more depth to know what is happening at the origin or minimum of the graph of our parabola and how or set of points are changing.
Now that we know this information we acknowledge that the line tangent to our parabola gives us the instantaneous velocity for every set of points of the parabola. Notice what happens when x0=-1 the line line tangent to f(x) is equal to zero which means that our velocity reaches zero and in respect to our car it had to bring it's velocity to a complete stop in order to change it's path.