Find the vertex, focus, and axis of symmetry of the parabola x^2-10x-8y+29 = 0.
==> I do it over and over and over, but I keep getting the same answers of (5,4) for the vertex, (5,6) for the focus, and x = 5 for the LOS, but according to this one website, everything except for the LOS is wrong. This is how I got the equation into vertex form:
x^2-10x+25 = 8y-29+25
(x-5)^2=8y-4
y = 1/8(x-5)^2+4
I don't see what I'm doing wrong -- can someone help me please? Thanks!! :)
Ohh never mind, I found my error. I feel kinda stupid now though lol
To find the vertex, focus, and axis of symmetry of a parabola, you need to convert the given equation into vertex form, which is of the form y = a(x-h)^2 + k.
Let's analyze your steps:
1. Start with: x^2-10x-8y+29 = 0
2. Rewrite the equation by completing the square for the x-terms: (x^2-10x+25) - 8y = 4
Notice that when you complete the square, you need to add and subtract the square of half the coefficient of the x-term, not just add. In this case, the coefficient is -10, so you need to add and subtract (-10/2)^2 = 25.
3. Simplify the equation: (x-5)^2 - 8y = 4
4. Rearrange it to y = form: 8y = (x-5)^2 - 4
Divide everything by 8 to make the coefficient of y 1/8.
y = 1/8(x-5)^2 - 1/2
So far, your calculations are all correct.
The equation is now in vertex form: y = a(x-h)^2 + k, where the vertex is the point (h, k).
From the equation, we can see that the vertex is (5, -1/2). So you calculated the vertex correctly.
To find the focus, you need to determine the value of a. In the vertex form, a determines the vertical stretch/compression of the parabola. The equation y = a(x-h)^2 + k is equivalent to (x-h)^2 = 4a(y-k). We can see that a = 1/8.
The focus of the parabola is given by the point (h, k+a), where a is the coefficient of y. In this case, the focus is (5, -1/2 + 1/8) = (5, 3/8).
Now, let's talk about the axis of symmetry (LOS). The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetric halves. Since the parabola is symmetric with respect to the line x=h, the axis of symmetry is x = 5.
Therefore, your calculations for the vertex, focus, and axis of symmetry are correct. It seems like either the website you mentioned has incorrect information, or there might be some confusion.
Double-check your calculations and consider seeking help from a math instructor if you encounter any further difficulties.