Identity The Translations Of The Parent Function F(×)=ײ That Give G(×)=(×-5)²-4
**Identity the translations of the parent function f(×)=ײ that give g(×)=(×-5)²-4**
Understanding the Parent Function
The parent function f(x) = x² is a quadratic function that represents a parabola with its vertex at the origin (0, 0). This function is the foundation for all quadratic functions, and it can be transformed in various ways to create new functions.
Translations of the Parent Function
When we talk about translations of the parent function, we are referring to the process of shifting the parent function up, down, left, or right, or reflecting it across the x-axis or y-axis. These transformations can be represented algebraically by changing the equation of the parent function.
Vertical Shifts
A vertical shift is a translation that moves the parent function up or down. If we add a constant to the parent function, it will shift the function up by that constant. If we subtract a constant from the parent function, it will shift the function down by that constant.
Horizontal Shifts
A horizontal shift is a translation that moves the parent function left or right. If we add a constant to the x-term of the parent function, it will shift the function left by that constant. If we subtract a constant from the x-term of the parent function, it will shift the function right by that constant.
Reflections
A reflection is a translation that flips the parent function across the x-axis or y-axis. If we multiply the parent function by -1, it will reflect the function across the x-axis. If we replace x with -x in the parent function, it will reflect the function across the y-axis.
Combining Translations
When we combine multiple translations, we can create a new function that is a result of all the individual translations. For example, if we shift the parent function up by 3 units, left by 2 units, and reflect it across the x-axis, the resulting function will be a combination of all these translations.
Example: g(x) = (x-5)² - 4
Now, let's apply the concept of translations to the given function g(x) = (x-5)² - 4. To identify the translations of the parent function f(x) = x² that give g(x), we need to analyze the equation of g(x).
Step 1: Identify the Horizontal Shift
The equation g(x) = (x-5)² - 4 has a term (x-5) inside the parentheses. This indicates that the parent function has been shifted 5 units to the right.
Step 2: Identify the Vertical Shift
The equation g(x) = (x-5)² - 4 also has a term -4 outside the parentheses. This indicates that the parent function has been shifted 4 units down.
Step 3: Identify the Reflection
There is no indication of a reflection in the equation g(x) = (x-5)² - 4.
Conclusion
In conclusion, the translations of the parent function f(x) = x² that give g(x) = (x-5)² - 4 are:
- A horizontal shift of 5 units to the right
- A vertical shift of 4 units down
These translations can be represented algebraically by changing the equation of the parent function.
Frequently Asked Questions
Q: What is the parent function?
A: The parent function is a quadratic function that represents a parabola with its vertex at the origin (0, 0). It is the foundation for all quadratic functions.
Q: What are the types of translations?
A: The types of translations are vertical shifts, horizontal shifts, and reflections.
Q: How do I identify the horizontal shift?
A: To identify the horizontal shift, look for a term inside the parentheses that has a constant added or subtracted from x.
Q: How do I identify the vertical shift?
A: To identify the vertical shift, look for a term outside the parentheses that has a constant added or subtracted from the function.
Q: How do I identify the reflection?
A: To identify the reflection, look for a term that has been multiplied by -1 or replaced with -x.
Q: Can I combine multiple translations?
A: Yes, you can combine multiple translations to create a new function that is a result of all the individual translations.
Q: How do I represent translations algebraically?
A: You can represent translations algebraically by changing the equation of the parent function.
Q: What is the equation of the parent function?
A: The equation of the parent function is f(x) = x².
Q: What is the equation of the given function?
A: The equation of the given function is g(x) = (x-5)² - 4.
Q: What are the translations of the parent function that give g(x)?
A: The translations of the parent function that give g(x) are a horizontal shift of 5 units to the right and a vertical shift of 4 units down.