Parabolas are a common curve defined by a set of rules. Parabolas have a specific axis of rotation, called the directrix, and a specific value for which the parabola opens, called the focus.
Parabolas can be rotated in either direction along their directrix. They can also be mirrored across their focus. When doing these operations, you will get new parabolas, but they will still meet the set of rules for parabolas.
Parabolic flight is one example of using the parabolic curve in aviation. Parabolic flight is when an aircraft flies straight up then down in such a way that it gains maximum speed and height due to gravity before returning back to where it took off. This is done in order to make the most use of available time and space before having to land.
This article will discuss how to derive the equation of a parabola with a focus at (−5, −5) and a directrix of Y = 7.
Calculate the focus and directrix
To calculate the focus and directrix of a parabola, you must first calculate the slope of the line that passes through the points on either side of the vertex.
The slope of the line passing through the points on either side of the vertex is called the negative reciprocity property. This property states that if a vertical line passes through two points on either side of a curve, then its slope is equal to -1 times the derivative of y at that point.
For our problem, we need to find out what -5 is divided by 7, or what the slope is. Then, we must find out which points we are looking for to find out where the focus and directrix are.
The focus is located five units away from (-5, −5) in either direction along the axis of symmetry.
Plug in X = −5 into f(x) = x2 − 5x + 7
To find the equation of the parabola with a focus at (−5, −5), or where the vertical axis of the parabola meets the horizontal axis, you must first find the variables for f(x) = x2 − 5x + 7.
Then, you must plug in −5 for x into this equation to get the new variable. In this case, you are substituting −5 for x to get y = −5. You then have to solve for y to get the focus of the parabola.
You can now plug in y = −5 and x = −5 into x2− 5x + 7 to find that the focus of this parabola is (0, 0).
Find the y-intercept
The next step is to find the y-intercept, which is where the graph crosses the y-axis. To do this, you need to find where the parabola equals 0.
To do this, subtract 7 from both sides of the equation. This gives you y = −5 + 7 = 2. So at 2, the graph crosses the y-axis. You can also say that the parabola drops 2 units below the y-axis.
This can help you identify points on the graph when solving problems using functions based on parabolas. For example, if a problem says something like “at x = 3,” then there is no point at 3 on the graph because it does not cross the x-axis.
Derive the equation of the parabola
To derive the equation of the parabola with a focus at (−5, −5) and a directrix of y = 7, first subtract 5 from both sides of the equation. Then, divide both sides by −5.
y = 7 − 5(x + 5)
y = 7 − 25 x
Next, simplify the right side by using the properties of equality to combine like terms and distribute 25 to each term in the denominator. Finally, reverse the order of the variables to get the complete equation.
Check by plotting points
Once you have derived the equation of the parabola, you can check by plotting points. Parabolas have two special points, called focal points.
The first focal point is located at the origin (0, 0) and the second focal point is located on the directrix. When plotting points, first plot the points on the graph using (0, 0) as the first coordinate and any integer value for the second coordinate.
Then, plot a point with a third coordinate of 7 and see if it is on the parabola. If it is not, then your equation is not correct! If it is, then plot all of the other points and see if they all line up with the curve of the parabola.
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