Find the quadratic function with vertex (1, -4) and passing through the point (2, -3).

Question

Find the quadratic function with the vertex (1, -4) and passing through the point (2, -3).

Answer 1

Vertex of a quadratic function, y=ax²+bx+c is given by x_v=-b/(2a);

y_v=c-b²/(4a); So we have:

 

-b/(2a)=1

c-b²/(4a)=-4

 

Third equation comes from the fact that our function passes through the point (2,-3).

 

4a+2b+c=-3

 

We get b=-2a from the first, c=-3-4a-2b=-3-4a+4a=-3 from the third equation. From the second equation we obtain:

 

-3-(-2a)²/(4a)=-4

 

a=1

 

Now b=-2a=-2.

 

Finally, we are ready to write our quadratic function:

 

Answer: x²-2x-3

 

It is easy to verify that this function indeed passes through the point (2, -3) and has a vertex at (1,-4)

Answer 2

Using vertex form,

y = a(x-1)^2-4

Plug in (2, -3),

-3 = a-4 => a = 1

Answer: y = (x-1)^2 - 4