write the equation of the parabola in vertex form. vertex (0,3), point (-4, -45)

Question

Write the equation of the parabola in vertex form.

vertex (0,3), point(-4, -45)

 

Answer 1

Recall that a parabola is the graph representing a quadratic equation, which is standard form is as follows:

     y = ax² + bx + c 

     where the value of 'a' determines whether the parabola opens upwards of downwards (i.e., the parabola opens upwards if a>0 and opens downwards if a<0) and the axis of symmetry is given by the line x = -b/(2a).

The vertex form of a parabola's (or a quadratic) equations is given by the following formula:

     y = a(x - h)² + k   ,   where (h, k) is the vertex and the axis of symmetry is given by the line  x = h.

Given that the vertex is at (0, 3), then ..... h = 0   and   k = 3 ..... thus,

     y = a(x - 0)² + 3

     y = a(x)² + 3

With a point at (-4, -45), then ..... x = -4   and   y = -45 ..... therefore,

     -45 = a(-4)² + 3

     -45 = a(16) + 3

Solve for a by first subtracting 3 from both sides of the equation then dividing both sides of the equation by 16:

     -45 - 3 = a(16) + 3 - 3

     -48 = a(16)

     -48/16 = 16a/16

     -3 = a

With a vertex (h, k) at (0, 3) and given that  a = -3, then the equation of this parabola in vertex form is as follows:

     y = a(x - h)² + k

     y = -3(x - 0)² + 3

     y = -3x² + 3