Find $g(x)$, Where $g(x)$ Is The Translation 1 Unit Right And 1 Unit Down Of \$f(x)=x^2$[/tex\].Write Your Answer In The Form $a(x - H)^2 + K$, Where $a$, $h$, And $k$ Are
Introduction
In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). When we translate a function, we are essentially moving its graph to a new position on the coordinate plane. In this article, we will explore how to find the translation of a given function, specifically the translation of $f(x) = x^2$ one unit to the right and one unit down.
Understanding Function Translation
When we translate a function, we are changing its position on the coordinate plane. There are two types of translations: horizontal and vertical. A horizontal translation moves the function's graph to the left or right, while a vertical translation moves the function's graph up or down.
Horizontal Translation
A horizontal translation of a function $f(x)$ by $h$ units to the right is given by $f(x - h)$. This means that we replace $x$ with $x - h$ in the original function.
Vertical Translation
A vertical translation of a function $f(x)$ by $k$ units down is given by $f(x) + k$. This means that we add $k$ to the original function.
Translation of $f(x) = x^2$
Now, let's apply the translation rules to the function $f(x) = x^2$. We want to translate it one unit to the right and one unit down.
Step 1: Horizontal Translation
To translate the function one unit to the right, we replace $x$ with $x - 1$ in the original function:
Step 2: Vertical Translation
To translate the function one unit down, we add $-1$ to the original function:
Simplifying the Expression
We can simplify the expression for $g(x)$ by expanding the squared term:
Standard Form
We want to write the expression for $g(x)$ in the standard form $a(x - h)^2 + k$, where $a$, $h$, and $k$ are constants.
Step 1: Completing the Square
To complete the square, we need to add and subtract $\left(\frac{b}{2}\right)^2$ inside the parentheses:
Step 2: Writing in Standard Form
Now, we can write the expression for $g(x)$ in the standard form:
Conclusion
In this article, we explored how to find the translation of a given function, specifically the translation of $f(x) = x^2$ one unit to the right and one unit down. We applied the translation rules to the original function and simplified the expression to write it in the standard form $a(x - h)^2 + k$. The final answer is:
g(x) = (x - 1)^2 - 1$<br/> **Translation of Functions: Q&A** ============================= **Introduction** --------------- In our previous article, we explored how to find the translation of a given function, specifically the translation of $f(x) = x^2$ one unit to the right and one unit down. In this article, we will answer some frequently asked questions about function translation. **Q&A** ------ ### Q: What is function translation? A: Function translation is the process of moving a function's graph to a new position on the coordinate plane. This can be done horizontally (left or right) or vertically (up or down). ### Q: How do I translate a function horizontally? A: To translate a function horizontally, you replace $x$ with $x - h$ in the original function, where $h$ is the number of units you want to translate the function to the right. ### Q: How do I translate a function vertically? A: To translate a function vertically, you add $k$ to the original function, where $k$ is the number of units you want to translate the function down. ### Q: What is the difference between a horizontal and vertical translation? A: A horizontal translation moves the function's graph to the left or right, while a vertical translation moves the function's graph up or down. ### Q: How do I write a translated function in the standard form $a(x - h)^2 + k$? A: To write a translated function in the standard form, you need to complete the square by adding and subtracting $\left(\frac{b}{2}\right)^2$ inside the parentheses. ### Q: What is the significance of the constants $a$, $h$, and $k$ in the standard form? A: The constants $a$, $h$, and $k$ represent the amplitude, horizontal shift, and vertical shift of the function, respectively. ### Q: Can I translate a function more than one unit? A: Yes, you can translate a function more than one unit. For example, to translate a function two units to the right and two units down, you would replace $x$ with $x - 2$ and add $-2$ to the original function. ### Q: How do I determine the direction of the translation? A: The direction of the translation is determined by the sign of the number you are adding or subtracting from the original function. If you are adding a positive number, the function will move up. If you are adding a negative number, the function will move down. If you are subtracting a positive number, the function will move left. If you are subtracting a negative number, the function will move right. ### Q: Can I translate a function that is not in the form $f(x) = x^2$? A: Yes, you can translate any function that is in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. ### Q: How do I graph a translated function? A: To graph a translated function, you can use the following steps: 1. Graph the original function. 2. Move the graph to the left or right by the specified number of units. 3. Move the graph up or down by the specified number of units. **Conclusion** ---------- In this article, we answered some frequently asked questions about function translation. We hope that this article has helped you to better understand the concept of function translation and how to apply it to different types of functions.