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Mrac [35]
3 years ago
8

What is the equation of the quadratic graph with a focus of (3, 6) and a directrix of y = 4? f(x) = one fourth (x − 3)2 + 1 f(x)

= one fourth (x − 3)2 + 5 f(x) = −one fourth (x − 2)2 + 5 f(x) = −one fourth (x − 2)2
Mathematics
1 answer:
mina [271]3 years ago
8 0
 Let point (x, y) be any point on the graph, than the distance between (x, y) and the focus (3, 6) is sqrt((x - 3)^2 + (y - 6)^2) and the distance between (x, y) and the directrix, y = 4 is |y - 4|

Thus sqrt((x - 3)^2 + (y - 6)^2) = |y - 4|
(x - 3)^2 + (y - 6)^2 = (y - 4)^2
x^2 - 6x + 9 + y^2 - 12y + 36 = y^2 - 8y + 16
x^2 - 6x + 29 = -8y + 12y = 4y
(x - 3)^2 + 20 = 4y
y = 1/4(x - 3)^2 + 5

Required answer is f(x) = one fourth (x - 3)^2 + 5
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Answer:

y=16x+1

Step-by-step explanation:

You want to find the equation for a line that passes through the two points:

(1,17) and (2,33).

First of all, remember what the equation of a line is:

y = mx+b

Where:

m is the slope, and

b is the y-intercept

First, let's find what m is, the slope of the line...

The slope of a line is a measure of how fast the line "goes up" or "goes down". A large slope means the line goes up or down really fast (a very steep line). Small slopes means the line isn't very steep. A slope of zero means the line has no steepness at all; it is perfectly horizontal.

For lines like these, the slope is always defined as "the change in y over the change in x" or, in equation form:

So what we need now are the two points you gave that the line passes through. Let's call the first point you gave, (1,17), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=1 and y1=17.

Also, let's call the second point you gave, (2,33), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=2 and y2=33.

Now, just plug the numbers into the formula for m above, like this:

m=  

33 - 17/  2 - 1

or...

m=  16/ 1

or...

m=16

So, we have the first piece to finding the equation of this line, and we can fill it into y=mx+b like this:

y=16x+b

Now, what about b, the y-intercept?

To find b, think about what your (x,y) points mean:

(1,17). When x of the line is 1, y of the line must be 17.

(2,33). When x of the line is 2, y of the line must be 33.

Because you said the line passes through each one of these two points, right?

Now, look at our line's equation so far: y=16x+b. b is what we want, the 16 is already set and x and y are just two "free variables" sitting there. We can plug anything we want in for x and y here, but we want the equation for the line that specfically passes through the two points (1,17) and (2,33).

So, why not plug in for x and y from one of our (x,y) points that we know the line passes through? This will allow us to solve for b for the particular line that passes through the two points you gave!.

You can use either (x,y) point you want..the answer will be the same:

(1,17). y=mx+b or 17=16 × 1+b, or solving for b: b=17-(16)(1). b=1.

(2,33). y=mx+b or 33=16 × 2+b, or solving for b: b=33-(16)(2). b=1.

See! In both cases we got the same value for b. And this completes our problem.

The equation of the line that passes through the points

(1,17) and (2,33)  is   y=16x+1

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Answer:

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Step-by-step explanation:

1.66/15.25=9.18

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