Given the graph, we are asked to find the equation of the parabola. This can be seen below.
Explanation
The equation of a parabola given in vertex form is
[tex]\begin{gathered} y=a(x-h)^2+k \\ \text{where (h,k) is the vertex of the parabola} \end{gathered}[/tex]
The vertex of a parabola is the point at the intersection of the parabola and its line of symmetry. In this case, the vertex is at the point (1,-2)
Therefore, we will substitute the vertex parameters into the equation of the parabola.
[tex]y=a(x-1)^2-2[/tex]
To get the constant "a" we will pick a point from the graph and substitute it into the formula.
If we pick a point, let say (-3,2) we will have
[tex]\begin{gathered} 2=a(-3-1)^2-2 \\ 2=a(-4)^2-2 \\ 2=16a-2 \\ 16a=4 \\ a=\frac{4}{16} \\ a=\frac{1}{4} \end{gathered}[/tex]
Therefore, the equation will become;
Answer:
[tex]y=\frac{1}{4}(x-1)^2-2[/tex]
