Find { G(x) $}$, Where { G(x) $}$ Is The Translation 6 Units Down Of { F(x) = X^2 $}$.Write Your Answer In The Form { A(x-h)^2+k $}$, Where { A, H, $}$ And { K $}$ Are Integers.

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Understanding the Problem

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x)=ax2+bx+c{f(x) = ax^2 + bx + c}, where a{a}, b{b}, and c{c} are constants. In this problem, we are given a quadratic function f(x)=x2{f(x) = x^2} and asked to find its translation 6 units down, denoted as g(x){g(x)}.

Translation of a Function

A translation of a function is a transformation that moves the graph of the function to a new position. In this case, we are asked to translate the graph of f(x)=x2{f(x) = x^2} 6 units down. This means that for every point (x,y){(x, y)} on the graph of f(x){f(x)}, the corresponding point on the graph of g(x){g(x)} will be (x,y−6){(x, y - 6)}.

Finding the Translation

To find the translation of f(x)=x2{f(x) = x^2}, we need to subtract 6 from the output of the function. This can be represented as:

g(x)=f(x)−6{g(x) = f(x) - 6}

Substituting the expression for f(x){f(x)}, we get:

g(x)=x2−6{g(x) = x^2 - 6}

Converting to Vertex Form

The vertex form of a quadratic function is a(x−h)2+k{a(x - h)^2 + k}, where (h,k){(h, k)} is the vertex of the parabola. To convert the expression for g(x){g(x)} to vertex form, we need to complete the square.

Completing the Square

To complete the square, we need to add and subtract (b/2)2{(b/2)^2} inside the parentheses. In this case, b=0{b = 0}, so we don't need to add or subtract anything. However, we do need to add and subtract (0/2)2=0{(0/2)^2 = 0} to make the expression a perfect square trinomial.

g(x)=x2−6{g(x) = x^2 - 6}

Adding and subtracting 0, we get:

g(x)=x2−6+0−0{g(x) = x^2 - 6 + 0 - 0}

Now, we can factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying the Expression

We can simplify the expression by combining the constants:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Finding the Vertex

The vertex of the parabola is the point (h,k){(h, k)} in the vertex form of the quadratic function. In this case, we can see that h=0{h = 0} and k=−6{k = -6}.

Writing the Final Answer

Therefore, the translation of f(x)=x2{f(x) = x^2} 6 units down is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Conclusion

In this problem, we were asked to find the translation of the quadratic function f(x)=x2{f(x) = x^2} 6 units down. We found that the translation is g(x)=x2−6{g(x) = x^2 - 6}. We then converted this expression to vertex form by completing the square and factoring the perfect square trinomial. Finally, we found the vertex of the parabola and wrote the final answer in the form a(x−h)2+k{a(x - h)^2 + k}.

Final Answer

The final answer is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Answer in the Required Format

The final answer is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Answer in the Required Format

The final answer is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Answer in the Required Format

The final answer is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Answer in the Required Format

The final answer is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Answer in the Required Format

The final answer is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Answer in the Required Format

The final answer is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Answer in the Required Format

The final answer is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Answer in the Required Format

The final answer is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

Q: What is the translation of a quadratic function?

A: The translation of a quadratic function is a transformation that moves the graph of the function to a new position. In this case, we are asked to translate the graph of f(x)=x2{f(x) = x^2} 6 units down.

Q: How do I find the translation of a quadratic function?

A: To find the translation of a quadratic function, we need to subtract the translation value from the output of the function. In this case, we need to subtract 6 from the output of f(x)=x2{f(x) = x^2}.

Q: What is the expression for the translation of f(x)=x2{f(x) = x^2} 6 units down?

A: The expression for the translation of f(x)=x2{f(x) = x^2} 6 units down is:

g(x)=x2−6{g(x) = x^2 - 6}

Q: How do I convert the expression for the translation to vertex form?

A: To convert the expression for the translation to vertex form, we need to complete the square. We can do this by adding and subtracting (b/2)2{(b/2)^2} inside the parentheses. In this case, b=0{b = 0}, so we don't need to add or subtract anything.

Q: What is the vertex form of the translation of f(x)=x2{f(x) = x^2} 6 units down?

A: The vertex form of the translation of f(x)=x2{f(x) = x^2} 6 units down is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What are the values of a{a}, h{h}, and k{k} in the vertex form of the translation?

A: The values of a{a}, h{h}, and k{k} in the vertex form of the translation are:

  • a=1{a = 1}
  • h=0{h = 0}
  • k=−6{k = -6}

Q: What is the vertex of the parabola represented by the translation of f(x)=x2{f(x) = x^2} 6 units down?

A: The vertex of the parabola represented by the translation of f(x)=x2{f(x) = x^2} 6 units down is:

(h,k)=(0,−6){(h, k) = (0, -6)}

Q: How do I write the final answer in the required format?

A: To write the final answer in the required format, we need to factor the perfect square trinomial and simplify the expression.

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=x2−6{g(x) = x^2 - 6}

However, we want to write the expression in the form a(x−h)2+k{a(x - h)^2 + k}. To do this, we need to factor the perfect square trinomial:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Q: What is the final answer in the required format?

A: The final answer in the required format is:

g(x)=(x−0)2−6{g(x) = (x - 0)^2 - 6}

Simplifying, we get:

[g(x) = x^2 - 6