We are to find the equation of the quadratic curve
By using the vertex method
The equation will be in the form
[tex]y=a(x-h)^2+k[/tex]
where, the vertex is at (h, k)
From the graph in the question
The vertex is at (-2, -3)
Therefore,
[tex]h=-2,k=-3[/tex]
Therefore the equation of the curve will become
[tex]\begin{gathered} y=a(x-(-2))^2+(-3) \\ y=a(x+2)^2-3 \end{gathered}[/tex]
Next, we need to find the value of a
Considering a point on the curve
We have the point (0, 1) on the curve
Hence we have
x = 0, y = 1
Substituting these values in the equation above we have
[tex]\begin{gathered} 1=a(0+2)^2-3 \\ 1=a(2)^2-3 \\ 1+3=4a \\ 4=4a \\ a=\frac{4}{4} \\ a=1 \end{gathered}[/tex]
Therefore, a = 1
Hence the equation of the curve will now become
[tex]\begin{gathered} y=1(x+2)^2-3 \\ y=(x+2)^2-3 \end{gathered}[/tex]
Therefore, the answer is
[tex]y=(x+2)^2-3[/tex]
