Explain how you can tell what kind of conic section a quadratic equation describes without graphing the equation.
Can you do this by writing the equation in standard form? (That's what my answer key says... but I don't know how/if it's actually right)
Yes, you can determine the type of conic section described by a quadratic equation by writing the equation in standard form. The standard form for a quadratic equation is:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
To identify the conic section, you need to analyze the coefficients A, B, and C. Here's how you can do it:
1. Circle: A, B, and C are all positive and equal,
A = C > 0, B = 0
2. Ellipse: A and C have the same sign and are not equal, B = 0,
A > 0, C > 0
3. Hyperbola: A and C have opposite signs, B = 0,
A > 0, C < 0 or A < 0, C > 0
4. Parabola: A or C is zero, but not both, and B^2 - 4AC = 0,
A = 0, C ≠ 0 or A ≠ 0, C = 0
It's important to note that the equation should be in standard form to correctly determine the conic section. If the equation is not in standard form, you can rearrange it by completing the square or factoring out any common factors.
To summarize, by writing the quadratic equation in standard form, you can determine the conic section by analyzing the coefficients.
Yes
like (x-h)^2= 4(a(y-k) is a parabola
(x-h)^2/a^2 + (y-k)^2/b^2 = 1 is an ellipse
if a^2=b^2 then circle of radius a^2=b^2=r^2
if - sign between, then ellipse turns to hyperbola etc