for the conic y=5x^2-40x+78 find an equation in standard form and its vertex, focus, and directrix
complete the square
y = 5(x^2 - 8x + ....) + 78
= 5(x^2 - 8x + 16 - 16) + 78
= 5( (x-4)^2 - 16) + 78
= 5(x-4)^2 - 80 + 78
= 5(x-4)^2 - 2
take over ....
What's my next step??
We could write our answer in the form
y+2 = 5(x-4)^2
take a look at
http://jwilson.coe.uga.edu/emt725/class/sarfaty/emt669/instructionalunit/parabolas/parabolas.html
half-way down the page at example 3
and follow the steps.
thank you!
To find the equation in standard form of the given conic, you need to complete the square for the quadratic expression. The standard form equation for a parabola is given by:
y = a(x - h)^2 + k
Where (h, k) represents the coordinates of the vertex.
Let's start by completing the square for the given equation y = 5x^2 - 40x + 78:
1. Divide the equation through by the common factor of 5:
y/5 = x^2 - 8x + 15.6
2. To complete the square, take half the coefficient of x (-8) and square it:
(-8/2)^2 = 16
3. Add the squared value to both sides of the equation:
y/5 + 16 = x^2 - 8x + 16 + 15.6
4. Simplify the right side of the equation:
y/5 + 16 = (x - 4)^2 + 31.6
5. Rearrange the equation to obtain the standard form:
(x - 4)^2 = 5(y/5 + 16 - 31.6)
(x - 4)^2 = 5(y/5 - 15.6)
Now, we have the equation in standard form: (x - 4)^2 = 5(y/5 - 15.6).
Comparing this equation with the standard form equation y = a(x - h)^2 + k, we can identify the vertex as (h, k) = (4, -15.6).
To find the focus and directrix of the parabola, we need to determine the value of the parameter 'p'. For a parabola of the form (x - h)^2 = 4p(y - k), the focus is given by (h, k + p) and the directrix is the horizontal line y = k - p.
From the standard form equation, we can see that 4p = 5. So, p = 5/4.
The vertex is (h, k) = (4, -15.6), so the focus coordinates can be obtained by adding p to the y-coordinate of the vertex: (4, -15.6 + 5/4). Simplifying, the focus is located at (4, -14.35).
Similarly, the directrix is a horizontal line located at y = -15.6 - 5/4. Simplifying, the directrix is y = -16.85.
Therefore, the equation of the given conic in standard form is:
(x - 4)^2 = 5(y/5 - 15.6)
The vertex is (4, -15.6), the focus is (4, -14.35), and the directrix is y = -16.85.