Question

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Answer 1

Answer:

$g(x) = x^{2} + 6\cdot x + 7$

Step-by-step explanation:

The blue parabola is only a translated version of the red parabola. The standard form of a vertical parabola centered at (h,k), that is, a parabola whose axis of symmetry is parallel to y-axis, is of the form:

$y - k = C\cdot (x-h)^{2}$

Where:

$h$, $k$ - Horizontal and vertical components of the vertex with respect to origin, dimensionless.

$C$ - Vertex constant, dimensionless. (If C > 0, then vertex is an absolute minimum, but if C < 0, then vertex is an absolute maximum).

Since both parabolas have absolute minima and it is told that have the same shape, the vertex constant of the blue parabola is:

$C = 1$

After a quick glance, the location of the vertex of the blue parabola with respect to the origin is:

$V(x,y) = (-3,-2)$

The standard form of the blue parabola is $y+2 = (x+3)^{2}$. Its expanded form is obtained after expanding the algebraic expression and clearing the independent variable (y):

$y + 2 = x^{2} +6\cdot x + 9$

$y = x^{2} + 6\cdot x + 7$

Then, the blue parabola is represented by the following equations:

$g(x) = x^{2} + 6\cdot x + 7$