Answer:
The function that represents the graph is [tex]y = -2 + \frac{1}{2}\cdot x^{2}[/tex].
Step-by-step explanation:
The graphic represents a vertical parabola, whose standard equation is:
[tex]y-k = C\cdot (x-h)^{2}[/tex]
Where:
[tex]x[/tex] - Independent variable, dimensionless.
[tex]y[/tex] - Depedent variable, dimensionless.
[tex]h[/tex] - Horizontal component of the vertex, dimensionless.
[tex]k[/tex] - Vertical component of the vertex, dimensionless.
[tex]C[/tex] - Vertex constant, dimensionless. Where [tex]C > 0[/tex] when vertex is an absolute minimum, otherwise it is an absolute maximum.
According to the figure, vertex is located in [tex](0, -2)[/tex]. Now we determine the vertex constant by using the following values in the standard equation:
[tex]x = -2[/tex], [tex]y = 0[/tex], [tex]h = 0[/tex], [tex]k = -2[/tex]
[tex]C = \frac{y-k}{(x-h)^{2}}[/tex]
[tex]C = \frac{0-(-2)}{(-2-0)^{2}}[/tex]
[tex]C = \frac{2}{4}[/tex]
[tex]C = \frac{1}{2}[/tex]
The function that represents the graph is [tex]y = -2 + \frac{1}{2}\cdot x^{2}[/tex].
