Find \[$ G(x) \$\], Where \[$ G(x) \$\] Is The Translation 5 Units To The Right Of \[$ F(x) = X^2 \$\].Write Your Answer In The Form \[$ A(x - H)^2 + K \$\], Where \[$ A, H, \$\] And \[$ K \$\] Are
Introduction
In mathematics, the translation of a function refers to the process of shifting the graph of the function to a new position on the coordinate plane. This can be done in various ways, including shifting the graph horizontally, vertically, or a combination of both. In this article, we will focus on finding the translation of a quadratic function, specifically the translation 5 units to the right of the function { f(x) = x^2 $}$.
Understanding the Original Function
The original function is given by { f(x) = x^2 $}$. This is a quadratic function in the form { f(x) = ax^2 + bx + c $}$, where { a = 1 $}$, { b = 0 $}$, and { c = 0 $}$. The graph of this function is a parabola that opens upwards, with its vertex at the origin (0, 0).
Translation of a Function
To translate a function, we need to understand the concept of horizontal and vertical shifts. A horizontal shift to the right or left is achieved by adding or subtracting a value from the input variable x, while a vertical shift up or down is achieved by adding or subtracting a value from the output variable f(x).
In this case, we are asked to find the translation 5 units to the right of the function { f(x) = x^2 $}$. This means that we need to shift the graph of the function 5 units to the right, which can be achieved by replacing x with (x - 5) in the original function.
Finding the Translation
To find the translation, we need to substitute (x - 5) for x in the original function { f(x) = x^2 $}$. This gives us:
{ g(x) = (x - 5)^2 $}$
Expanding the squared term, we get:
{ g(x) = x^2 - 10x + 25 $}$
This is the translation 5 units to the right of the original function { f(x) = x^2 $}$.
Writing the Translation in Vertex Form
The translation can also be written in vertex form, which is given by { a(x - h)^2 + k $}$, where { a, h, $}$ and { k $}$ are constants. To write the translation in vertex form, we need to complete the square.
Starting with the expanded form of the translation:
{ g(x) = x^2 - 10x + 25 $}$
We can complete the square by adding and subtracting the square of half the coefficient of x. In this case, the coefficient of x is -10, so half of it is -5, and its square is 25.
Adding and subtracting 25 inside the parentheses, we get:
{ g(x) = (x^2 - 10x + 25) - 25 $}$
Simplifying, we get:
{ g(x) = (x - 5)^2 - 25 $}$
This is the translation 5 units to the right of the original function { f(x) = x^2 $}$, written in vertex form.
Conclusion
In this article, we have discussed the concept of translating a quadratic function, specifically the translation 5 units to the right of the function { f(x) = x^2 $}$. We have shown how to find the translation by substituting (x - 5) for x in the original function, and how to write the translation in vertex form by completing the square. The translation is given by { g(x) = (x - 5)^2 - 25 $}$, which is the standard form of a quadratic function in vertex form.
Example Problems
- Find the translation 3 units to the right of the function { f(x) = x^2 + 2x + 1 $}$.
- Find the translation 2 units to the left of the function { f(x) = x^2 - 4x + 3 $}$.
- Find the translation 4 units up of the function { f(x) = x^2 - 2x + 1 $}$.
Solutions
- To find the translation 3 units to the right of the function { f(x) = x^2 + 2x + 1 $}$, we need to substitute (x - 3) for x in the original function.
{ g(x) = (x - 3)^2 + 2(x - 3) + 1 $}$
Expanding the squared term and simplifying, we get:
{ g(x) = x^2 - 6x + 9 + 2x - 6 + 1 $}$
Simplifying further, we get:
{ g(x) = x^2 - 4x + 4 $}$
This is the translation 3 units to the right of the original function.
- To find the translation 2 units to the left of the function { f(x) = x^2 - 4x + 3 $}$, we need to substitute (x + 2) for x in the original function.
{ g(x) = (x + 2)^2 - 4(x + 2) + 3 $}$
Expanding the squared term and simplifying, we get:
{ g(x) = x^2 + 4x + 4 - 4x - 8 + 3 $}$
Simplifying further, we get:
{ g(x) = x^2 - 1 $}$
This is the translation 2 units to the left of the original function.
- To find the translation 4 units up of the function { f(x) = x^2 - 2x + 1 $}$, we need to add 4 to the original function.
{ g(x) = x^2 - 2x + 1 + 4 $}$
Simplifying, we get:
{ g(x) = x^2 - 2x + 5 $}$
This is the translation 4 units up of the original function.
Final Answer
Introduction
In our previous article, we discussed the concept of translating quadratic functions, specifically the translation 5 units to the right of the function { f(x) = x^2 $}$. We showed how to find the translation by substituting (x - 5) for x in the original function, and how to write the translation in vertex form by completing the square. In this article, we will answer some frequently asked questions about translating quadratic functions.
Q: What is the difference between a horizontal shift and a vertical shift?
A: A horizontal shift is a change in the x-coordinate of the graph of a function, while a vertical shift is a change in the y-coordinate of the graph of a function. In the case of a quadratic function, a horizontal shift can be achieved by adding or subtracting a value from the input variable x, while a vertical shift can be achieved by adding or subtracting a value from the output variable f(x).
Q: How do I find the translation of a quadratic function?
A: To find the translation of a quadratic function, you need to substitute (x - h) for x in the original function, where h is the value of the horizontal shift. For example, if you want to find the translation 3 units to the right of the function { f(x) = x^2 + 2x + 1 $}$, you would substitute (x - 3) for x in the original function.
Q: What is the vertex form of a quadratic function?
A: The vertex form of a quadratic function is given by { a(x - h)^2 + k $}$, where { a, h, $}$ and { k $}$ are constants. The vertex form is useful for writing quadratic functions in a way that makes it easy to identify the vertex of the parabola.
Q: How do I complete the square to write a quadratic function in vertex form?
A: To complete the square, you need to add and subtract the square of half the coefficient of x inside the parentheses. For example, if you have the quadratic function { f(x) = x^2 + 4x + 4 $}$, you would add and subtract 4 inside the parentheses to get:
{ f(x) = (x^2 + 4x + 4) - 4 $}$
Simplifying, you get:
{ f(x) = (x + 2)^2 - 4 $}$
This is the vertex form of the quadratic function.
Q: What is the significance of the vertex of a parabola?
A: The vertex of a parabola is the point where the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on whether the parabola opens upwards or downwards.
Q: How do I find the vertex of a parabola?
A: To find the vertex of a parabola, you need to write the quadratic function in vertex form. The vertex is then given by the values of h and k in the vertex form.
Q: Can I use the vertex form to graph a quadratic function?
A: Yes, you can use the vertex form to graph a quadratic function. The vertex form makes it easy to identify the vertex of the parabola, which is the key to graphing the parabola.
Q: What are some common mistakes to avoid when translating quadratic functions?
A: Some common mistakes to avoid when translating quadratic functions include:
- Not substituting (x - h) for x in the original function when finding the translation.
- Not completing the square correctly when writing a quadratic function in vertex form.
- Not identifying the vertex of the parabola correctly.
Conclusion
In this article, we have answered some frequently asked questions about translating quadratic functions. We have discussed the concept of horizontal and vertical shifts, how to find the translation of a quadratic function, and how to write a quadratic function in vertex form by completing the square. We have also discussed the significance of the vertex of a parabola and how to find it. By following the tips and avoiding common mistakes, you can master the art of translating quadratic functions.
Example Problems
- Find the translation 2 units to the left of the function { f(x) = x^2 + 3x + 2 $}$.
- Find the translation 4 units up of the function { f(x) = x^2 - 2x + 1 $}$.
- Write the function { f(x) = x^2 + 4x + 4 $}$ in vertex form by completing the square.
Solutions
- To find the translation 2 units to the left of the function { f(x) = x^2 + 3x + 2 $}$, we need to substitute (x + 2) for x in the original function.
{ g(x) = (x + 2)^2 + 3(x + 2) + 2 $}$
Expanding the squared term and simplifying, we get:
{ g(x) = x^2 + 4x + 4 + 3x + 6 + 2 $}$
Simplifying further, we get:
{ g(x) = x^2 + 7x + 12 $}$
This is the translation 2 units to the left of the original function.
- To find the translation 4 units up of the function { f(x) = x^2 - 2x + 1 $}$, we need to add 4 to the original function.
{ g(x) = x^2 - 2x + 1 + 4 $}$
Simplifying, we get:
{ g(x) = x^2 - 2x + 5 $}$
This is the translation 4 units up of the original function.
- To write the function { f(x) = x^2 + 4x + 4 $}$ in vertex form by completing the square, we need to add and subtract 4 inside the parentheses.
{ f(x) = (x^2 + 4x + 4) - 4 $}$
Simplifying, we get:
{ f(x) = (x + 2)^2 - 4 $}$
This is the vertex form of the function.
Final Answer
The final answer is { g(x) = (x - 5)^2 - 25 $}$.