Find \[$ G(x) \$\], Where \[$ G(x) \$\] Is The Translation 4 Units Left And 2 Units Down Of \[$ F(x) = X^2 \$\].Write Your Answer In The Form \[$ A(x-h)^2+k \$\], Where \[$ A, H \$\], And \[$ K \$\] Are
Understanding Function Translations
In mathematics, function translations refer to the process of shifting a function's graph to a new position on the coordinate plane. This can be achieved by modifying the function's equation to reflect the desired translation. In this article, we will explore how to find the translation of a given function, specifically the translation 4 units left and 2 units down of the function { f(x) = x^2 $}$.
The Original Function: f(x)
The original function, { f(x) = x^2 $}$, is a quadratic function that represents a parabola opening upwards. The equation of this function can be written in the form { f(x) = a(x-h)^2+k $}$, where { a $}$ is the coefficient of the squared term, { h $}$ is the horizontal shift, and { k $}$ is the vertical shift.
For the function { f(x) = x^2 $}$, we have { a = 1 $}$, { h = 0 $}$, and { k = 0 $}$. This means that the original function is a standard quadratic function with no horizontal or vertical shifts.
Translation 4 Units Left and 2 Units Down
To translate the function 4 units left and 2 units down, we need to modify the equation of the function to reflect this new position. The horizontal shift of 4 units left means that we need to subtract 4 from the x-coordinate of the function. The vertical shift of 2 units down means that we need to subtract 2 from the y-coordinate of the function.
Finding g(x)
Using the information above, we can write the equation of the translated function, { g(x) $}$, as:
{ g(x) = f(x-4) - 2 $}$
Substituting the equation of the original function, { f(x) = x^2 $}$, into this equation, we get:
{ g(x) = (x-4)^2 - 2 $}$
Simplifying the Equation
To simplify the equation, we can expand the squared term and combine like terms:
{ g(x) = x^2 - 8x + 16 - 2 $}$
{ g(x) = x^2 - 8x + 14 $}$
Writing in the Form (x-h)^2+k
To write the equation in the form { a(x-h)^2+k $}$, we need to complete the square. We can do this by adding and subtracting the square of half the coefficient of the x-term:
{ g(x) = x^2 - 8x + 16 - 2 + 16 - 16 $}$
{ g(x) = (x-4)^2 - 2 + 16 - 16 $}$
{ g(x) = (x-4)^2 - 2 $}$
Conclusion
In this article, we have explored how to find the translation of a given function, specifically the translation 4 units left and 2 units down of the function { f(x) = x^2 $}$. We have shown that the equation of the translated function, { g(x) $}$, can be written in the form { a(x-h)^2+k $}$, where { a = 1 $}$, { h = 4 $}$, and { k = -2 $}$. This demonstrates the importance of understanding function translations in mathematics and how they can be used to analyze and solve problems involving functions.
Example Problems
- Find the translation of the function { f(x) = x^2 + 3 $}$ 2 units up and 1 unit right.
- Find the equation of the function { g(x) $}$ that is the translation 3 units left and 1 unit down of the function { f(x) = x^2 + 2x $}$.
- Find the equation of the function { g(x) $}$ that is the translation 2 units up and 1 unit right of the function { f(x) = x^2 - 4x $}$.
Solutions
- The equation of the translated function is { g(x) = (x+1)^2 + 5 $}$.
- The equation of the translated function is { g(x) = (x+3)^2 + 2x - 1 $}$.
- The equation of the translated function is { g(x) = (x-1)^2 + 4x + 2 $}$.
Tips and Tricks
- When translating a function, make sure to modify the equation of the function to reflect the desired translation.
- Use the formula { g(x) = f(x-h) + k $}$ to find the equation of the translated function.
- When writing the equation in the form { a(x-h)^2+k $}$, make sure to complete the square to simplify the equation.
Conclusion
In conclusion, function translations are an important concept in mathematics that can be used to analyze and solve problems involving functions. By understanding how to find the translation of a given function, we can write the equation of the translated function in the form { a(x-h)^2+k $}$. This demonstrates the importance of understanding function translations and how they can be used to solve problems in mathematics.
Understanding Function Translations
In our previous article, we explored how to find the translation of a given function, specifically the translation 4 units left and 2 units down of the function { f(x) = x^2 $}$. In this article, we will answer some frequently asked questions about function translations.
Q: What is a function translation?
A: A function translation is the process of shifting a function's graph to a new position on the coordinate plane. This can be achieved by modifying the function's equation to reflect the desired translation.
Q: What are the different types of function translations?
A: There are two main types of function translations: horizontal shifts and vertical shifts. Horizontal shifts involve moving the function's graph left or right, while vertical shifts involve moving the function's graph up or down.
Q: How do I find the equation of a translated function?
A: To find the equation of a translated function, you need to modify the equation of the original function to reflect the desired translation. This can be done by adding or subtracting a constant to the x-coordinate or y-coordinate of the function.
Q: What is the formula for finding the equation of a translated function?
A: The formula for finding the equation of a translated function is { g(x) = f(x-h) + k $}$, where { h $}$ is the horizontal shift and { k $}$ is the vertical shift.
Q: How do I write the equation of a translated function in the form { a(x-h)^2+k $}$?
A: To write the equation of a translated function in the form { a(x-h)^2+k $}$, you need to complete the square. This involves adding and subtracting the square of half the coefficient of the x-term.
Q: What is the importance of understanding function translations?
A: Understanding function translations is important because it allows you to analyze and solve problems involving functions. By understanding how to find the translation of a given function, you can write the equation of the translated function in the form { a(x-h)^2+k $}$.
Q: Can you give an example of a function translation?
A: Yes, consider the function { f(x) = x^2 $}$. If we want to translate this function 2 units up and 1 unit right, we would get the equation { g(x) = (x-1)^2 + 2 $}$.
Q: How do I determine the horizontal and vertical shifts of a function?
A: To determine the horizontal and vertical shifts of a function, you need to look at the equation of the function. If the equation is in the form { f(x) = a(x-h)^2+k $}$, then the horizontal shift is { h $}$ and the vertical shift is { k $}$.
Q: Can you give an example of a function with a horizontal shift?
A: Yes, consider the function { f(x) = (x-2)^2 $}$. This function has a horizontal shift of 2 units to the right.
Q: Can you give an example of a function with a vertical shift?
A: Yes, consider the function { f(x) = x^2 + 3 $}$. This function has a vertical shift of 3 units up.
Q: How do I determine the equation of a function with a horizontal shift?
A: To determine the equation of a function with a horizontal shift, you need to modify the equation of the original function to reflect the desired horizontal shift. This can be done by adding or subtracting a constant to the x-coordinate of the function.
Q: How do I determine the equation of a function with a vertical shift?
A: To determine the equation of a function with a vertical shift, you need to modify the equation of the original function to reflect the desired vertical shift. This can be done by adding or subtracting a constant to the y-coordinate of the function.
Conclusion
In conclusion, function translations are an important concept in mathematics that can be used to analyze and solve problems involving functions. By understanding how to find the translation of a given function, you can write the equation of the translated function in the form { a(x-h)^2+k $}$. This demonstrates the importance of understanding function translations and how they can be used to solve problems in mathematics.