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Answer:
(x²-10x+33)/(-8) = y
Step-by-step explanation:
The distance between any point on a parabola from both its focus and directrix are the same.
Let's say we have a point (x,y) on the parabola. We can then say that using the distance formula,
is the distance between (x,y) and the focus. Similarly, the distance between (x,y) and the directrix is |y-1| (I use absolute value here because distance is always positive). We can find this equation by taking the shortest distance from the point to the line. Because the closest point to the line will be the same x value as the point itself, the distance is simply the distance between the y value of the point and the y value of the directrix.
Equating the two equations given, we have
square both sides to get
(x-5)²+(y+3)²=(y-1)²
expand the y components
(x-5)² + y²+6y+9 = y²-2y+1
subtract y²+6y+9 from both sides
(x-5)² = -8y - 8
expand the x components
x²-10x+25 = -8y - 8
add 8 to both sides to isolate the -8y
x²-10x+33 = -8y
divide both sides by -8 to isolate y
(x²-10x+33)/(-8) = y
The equation is a function
Explanation:
Given that the graph contains the coordinates ,
,
and
We need to determine whether the equation is a function.
To determine the equation is a function, let us substitute the coordinates in the equation and check whether it satisfies the equation.
Let us substitute the coordinate in the equation
Thus, we have,
Thus, the coordinate satisfies the equation
Substituting the coordinate in the equation, we have,
Thus, the coordinate satisfies the equation
Substituting the coordinate in the equation
Thus, the coordinate satisfies the equation
Substituting the coordinate in the equation, we get,
Thus, the coordinate satisfies the equation
Hence, the coordinates ,
,
and
satisfies the equation
Thus, the equation is a function.