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how to solve a conic section problem

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My answer assumes that the (y+1) term is squared and that the -3, is subtracted from the product of 1/6 times (y+1)^2.  I'm going to write this equation as x = (1/6)(y+1)^2 - 3

If that is wrong, feel free to email or post a clarification.

To classify a conic section, the first question to ask is whether just one, or both variables are squared, i.e. is it x^2, y^2, or both.  If only one variable is squared, it will be a parabola.  If both are squared, it will be a circle/ellipse or a hyperbola.  So in this case we have a parabola, which looks like a half-pipe.

Once you have decided what kind of conic, the next question is does it open up or down or left or right.  Generally parabolas are up like a "U", down like an "n", right like a "<" or left like a ">".  This is determined by which variable is squared (x or y) and the sign (+ or -) in front of the squared term.
Up/Down means that x is squared; and Left/Right means that y is squared; it will be down or left if there is a minus (-) sign in front of the squared term.  In your case y is squared and it does not have a minus sign outside the parentheses, so it will open to the right.

While you can plot the graph by interpreting the (1/6), (+1), and (-3) terms directly, I recommend when starting out that you try another approach.  First plot some key points where both x and y will have clear, preferably integer (i.e. whole number) values and recognize that a parabola is symmetrical around its smallest or largest value.  In this case, what will x equal

(a) when y=-1, x=??  This is a good choice because it will eliminate the (1/6) term (-1+1)=0

(b) when y=0, x=??  Trying +/1 one unit from eliminating the (1/6) term is often useful

(c) when y=-2, x=??

(d) when y=5, x=??  Trying 5, will give us 6 squared, which will be an integer when multiplied by 1/6

(e) when y=-7, x=??

If you can plot these 5 points, you can graph the < shape you should get.

I hope this helps.  John