To find the coordinates of the point where the curve cuts the y-axis, we set $x = 0$ because any point on the y-axis has an $x$-coordinate of 0.

The equation of the curve is: $y = x^2 + 2x - 3$

Substitute $x = 0$ into the equation: $y = (0)^2 + 2(0) - 3$ $y = -3$

So, the coordinates of point $A$ where the curve cuts the y-axis are $(0, -3)$.

Let me know if you'd like more details, or if you have any other questions!

Here are 5 related questions you can explore:

1. What is the vertex of the parabola represented by the equation $y = x^2 + 2x - 3$?
2. How do you find the x-intercepts (roots) of the equation $y = x^2 + 2x - 3$?
3. What is the axis of symmetry of the parabola given by $y = x^2 + 2x - 3$?
4. How do you complete the square for the equation $y = x^2 + 2x - 3$?
5. What are the properties of parabolas in standard form $y = ax^2 + bx + c$?

Tip: When solving for the y-intercept of any quadratic equation, always substitute $x = 0$ into the equation, as this will give the point where the curve intersects the y-axis.