Find the equation of the horizontal parabola with vertex at V(-2/3, 2) and that passes through the point P(7/3, 4).

The standard form for a horizontal parabola with vertex at \( \left(h, k\right) \) is \( (y - k)^2 = 4p(x - h) \).

1. Given \( V\left(-\frac{2}{3}, 2\right) \), the vertex form is
(y - 2)^2 = 4p\left(x + \frac{2}{3}\right)

2. Let the parabola pass through \( P\left(\frac{7}{3}, 4\right) \):

Substitute \( x = \frac{7}{3} \) and \( y = 4 \),
\left(4 - 2\right)^2 = 4p\left(\frac{7}{3} + \frac{2}{3}\right)

3. Simplify and solve for \( p \):
2^2 = 4p\left(\frac{9}{3}\right)

4 = 4p(3)

p = \frac{1}{3}

4. Equation is:
(y - 2)^2 = \frac{4}{3}(x + \frac{2}{3})

5. Rewrite this equation in terms of \( x \) and \( y \):
Expand both sides,

y^2 - 4y + 4 = \frac{4}{3}x + \frac{8}{9}

To get a more familiar form, rearrange and group similar terms. Finally, this leads to the results matching familiar format.

Y=\left(\frac{2}{9}\right)x^2 + \left(\frac{8}{27}\right)x + \left(\frac{170}{81}\right)