Find \[$ G(x) \$\], Where \[$ G(x) \$\] Is The Translation 7 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.\[$
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) = ax^2 + bx + c $}$, where { a, b, $}$ and { c $}$ are constants. In this problem, we are given a quadratic function { f(x) = x^2 $}$ and asked to find its translation 7 units down, denoted as { 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 translating the graph of { f(x) = x^2 $}$ 7 units down. This means that for every point { (x, y) $}$ on the graph of { f(x) = x^2 $}$, the corresponding point on the graph of { g(x) $}$ will be { (x, y - 7) $}$.
Finding the Translation
To find the translation of { f(x) = x^2 $}$ 7 units down, we need to subtract 7 from the y-coordinate of each point on the graph. This can be represented algebraically as { g(x) = f(x) - 7 $}$. Substituting the expression for { f(x) = x^2 $}$, we get { g(x) = x^2 - 7 $}$.
Converting to Vertex Form
The vertex form of a quadratic function is { a(x-h)^2 + k $}$, where { a, h, $}$ and { k $}$ are integers. To convert the expression { g(x) = x^2 - 7 $}$ to vertex form, we need to complete the square. This involves rewriting the expression as a perfect square trinomial.
Completing the Square
To complete the square, we need to add and subtract { (b/2)^2 $}$ inside the parentheses. In this case, { b = 0 $}$, so we don't need to add or subtract anything. However, we do need to add and subtract { (0/2)^2 = 0 $}$ to make the expression a perfect square trinomial.
Finding the Vertex Form
After completing the square, we get { g(x) = (x-0)^2 - 7 $}$. This can be rewritten as { g(x) = (x-0)^2 - 7 $}$. Comparing this with the vertex form { a(x-h)^2 + k $}$, we can see that { a = 1 $}$, { h = 0 $}$, and { k = -7 $}$.
Conclusion
In conclusion, the translation of { f(x) = x^2 $}$ 7 units down is { g(x) = (x-0)^2 - 7 $}$. This can be rewritten as { g(x) = x^2 - 7 $}$. The vertex form of this function is { g(x) = (x-0)^2 - 7 $}$, where { a = 1 $}$, { h = 0 $}$, and { k = -7 $}$.
Final Answer
Frequently Asked Questions
In this article, we will answer some frequently asked questions related to finding the translation of a quadratic function.
Q: What is the translation of a function?
A: The translation of a function is a transformation that moves the graph of the function to a new position. In this case, we are translating the graph of { f(x) = x^2 $}$ 7 units down.
Q: How do I find the translation of a quadratic function?
A: To find the translation of a quadratic function, you need to subtract the translation value from the y-coordinate of each point on the graph. In this case, we subtract 7 from the y-coordinate of each point on the graph of { f(x) = x^2 $}$.
Q: What is the vertex form of a quadratic function?
A: The vertex form of a quadratic function is { a(x-h)^2 + k $}$, where { a, h, $}$ and { k $}$ are integers.
Q: How do I convert a quadratic function to vertex form?
A: To convert a quadratic function to vertex form, you need to complete the square. This involves rewriting the expression as a perfect square trinomial.
Q: What is completing the square?
A: Completing the square is a process of rewriting a quadratic expression as a perfect square trinomial. This involves adding and subtracting a constant value inside the parentheses.
Q: How do I find the vertex form of a quadratic function?
A: To find the vertex form of a quadratic function, you need to complete the square and rewrite the expression as a perfect square trinomial.
Q: What are the values of a, h, and k in the vertex form of a quadratic function?
A: The values of { a, h, $}$ and { k $}$ in the vertex form of a quadratic function depend on the specific function. In this case, { a = 1 $}$, { h = 0 $}$, and { k = -7 $}$.
Q: How do I use the vertex form of a quadratic function?
A: The vertex form of a quadratic function can be used to find the vertex of the parabola, which is the point where the parabola changes direction.
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola.
Q: How do I find the vertex of a parabola?
A: To find the vertex of a parabola, you need to use the vertex form of the quadratic function and find the values of { h $}$ and { k $}$.
Conclusion
In conclusion, finding the translation of a quadratic function involves subtracting the translation value from the y-coordinate of each point on the graph. The vertex form of a quadratic function is { a(x-h)^2 + k $}$, where { a, h, $}$ and { k $}$ are integers. Completing the square is a process of rewriting a quadratic expression as a perfect square trinomial. The vertex form of a quadratic function can be used to find the vertex of the parabola.
Final Answer
The final answer is { g(x) = (x-0)^2 - 7 $}$.