Question

Write an equation of the parabola that passes through the points at (-3,7), has its vertex at (-1,3), and opens to the left?

Answer 1

Start with the equation \(y = ax^2\) of a general parabola that opens upward and has its vertex at the origin.

Answer 2

Sorry, opens vertically. It might open downward.

Answer 3

We want it to open to the left so we interchange \(x\) and \(y\): \(x = ay^2\).

Answer 4

We want its vertex to be \((-1, 3)\) so we translate it by replacing \(x\) with \(x+1\) and \(y\) with \(y-3\): \(x+1=a(y-3)^2\).

Answer 5

We know that (-3, 7) satisfies the equation of our parabola: \[(-3)+1=a((7)-3)^2\\-2 = 16a\\a = -1/8\]
In conclusion, the equation is \[x+1=-(y-3)^2/8\]

Answer 6

Thank you! I

Answer 7

You're welcome!