Question

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the graph above is a transformation of the function ( x^2 ).
give the function in the graph above.
( g(x) = )
question help: video

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a quadratic function is $g(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola.

Step2: Identify vertex from graph

From the graph, the vertex (highest point) is at $(2, -3)$. So $h=2$, $k=-3$. Substitute into the form:
$g(x)=a(x-2)^2-3$

Step3: Solve for stretch factor $a$

Pick a point on the parabola, e.g., the x-intercept $(1, -6)$. Substitute $x=1$, $g(x)=-6$ into the equation:
$-6=a(1-2)^2-3$
Simplify:
$-6=a(1)-3$
$a=-6+3=-3$

Step4: Write final function

Substitute $a=-3$ back into the vertex form:
$g(x)=-3(x-2)^2-3$

Answer:

$g(x) = -3(x-2)^2 - 3$