The equation of a parabola with apex at the point (-1; -3) passing through the point (1; 1) has the form

The parabola equation has the form:

y = ax² + bx + c.

Let’s find the coefficients in this equation.

1) The vertex of the parabola is found by the formulas:

x0 = -b / 2a;

y0 = y (x0).

2) (-1; -3) is the vertex of the parabola, which means:

– 1 = – b / 2a;

2a = b.

The parabola equation takes the form:

y = ax² + 2ax + c.

3) Now let’s compose a system of equations by substituting the coordinates of these points into this equation:

-3 = a (-1) ² + 2a (-1) + c;
1 = a1² + 2a · 1 + c.

-3 = a – 2a + c;
1 = a + 2a + c.

c – a = -3;
c + 3a = 1.

c = a – 3;
c + 3a = 1.

a – 3 + 3a = 1;

4a = 1 + 3;

4a = 4;

a = 4: 4;

a = 1

c = a – 3 = 1 – 3 = -2.

b = 2a = 2.

Hence the original equation of the parabola:

y = x² + 2x – 2.