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The graph of g is a horizontal shrink by a factor of 1/2 and a translation 1 unit down, followed by a reflection in the x-axis of the graph of f(x)=(x+6)^2+3. Write a rule for g. Then identify the vertex.

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Answer:

steps below

Step-by-step explanation:

f(x)=(x+6)²+3  ... Gray graph

horizontal shrink by a factor of 1/2: f(2x) = (2x+6)²+3 ... green

translation 1 unit down: f(2x)-1 = (2x+6)²+2   ... purple

reflection in the x-axis: g(x) = - ((2x+6)²+2) = - (2x+6)²-2 = -4(x+3) - 2  .. Red

parabola equation: y = a(x-h)² + k    (h,k): vertex

vertex (-3 , -2)

Ver imagen lynnkc2000
MrRoyal

The rule of function g(x) is [tex]g(x) = -4(x + 3)^2 -2[/tex], and the vertex of the function is (-3,-2)

The equation of the function is given as:

[tex]f(x) = (x + 6)^2 + 3[/tex]

The rule of shrinking the function by a factor of 1/2 is:

[tex](x,y) \to (2x,y)[/tex]

So, we have:

[tex]f'(x) = (2x + 6)^2 + 3[/tex]

When the function is translated down by 1 unit, the rule of the transformation is:

[tex](x,y) \to (x,y-1)[/tex]

So, we have:

[tex]f"(x) = (2x + 6)^2 + 3-1[/tex]

[tex]f'"x) = (2x + 6)^2 + 2[/tex]

Lastly, the function is reflected across the x-axis,

The rule of this transformation is:

[tex](x,y) \to (x-y)[/tex]

So, we have:

[tex]g(x) = -(2x + 6)^2 -2[/tex]

Factor out 2

[tex]g(x) = -(2(x + 3))^2 -2[/tex]

Expand

[tex]g(x) = -4(x + 3)^2 -2[/tex]

The above represents the rule of function g(x), and the vertex of the function is (-3,-2)

Read more about function transformation at:

https://brainly.com/question/1548871

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