Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
Let f(x) = 4vI.
If g(a) is the graph of f(x) shifted down 2 units and left 4 units, write a formula for g(x).
![Title: Transforming Functions
Description: Understanding how transformations affect the graph of a function.
---
**Example Problem:**
Let \( f(x) = 4\sqrt{x} \).
If \( g(x) \) is the graph of \( f(x) \) shifted down 2 units and left 4 units, write a formula for \( g(x) \).
**Solution:**
To shift a function down by a certain number of units, you subtract that number from the function. For shifting down 2 units, the transformation will be:
\[ f(x) - 2 \]
To shift a function to the left by a certain number of units, you replace \( x \) with \( x + \text{that number} \). For shifting left 4 units, the transformation will be:
\[ f(x + 4) \]
Combining both transformations, we get:
\[ g(x) = f(x + 4) - 2 \]
Since \( f(x) = 4\sqrt{x} \), we substitute \( x + 4 \) for \( x \):
\[ f(x + 4) = 4\sqrt{x + 4} \]
Thus, applying the downward shift:
\[ g(x) = 4\sqrt{x + 4} - 2 \]
Therefore, the formula for \( g(x) \) is:
\[ g(x) = 4\sqrt{x + 4} - 2 \]
**Exercise:**
Enter the final formula for \( g(x) \):
\[ \boxed{g(x) = 4\sqrt{x + 4} - 2} \]
*Note: Enter \( \sqrt{x} \) as sqrt(x).*
[A button to "Add Work"]
[A button to proceed to the "Next Question"]
---
By following the steps outlined, students can better understand how to apply transformations to the graph of a function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f973d2e-ff2e-4b11-9052-66f4bc90f91c%2Fb754ac3a-3e43-4bdd-b5a2-ca920e885679%2Fg7pj4nj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Transforming Functions
Description: Understanding how transformations affect the graph of a function.
---
**Example Problem:**
Let \( f(x) = 4\sqrt{x} \).
If \( g(x) \) is the graph of \( f(x) \) shifted down 2 units and left 4 units, write a formula for \( g(x) \).
**Solution:**
To shift a function down by a certain number of units, you subtract that number from the function. For shifting down 2 units, the transformation will be:
\[ f(x) - 2 \]
To shift a function to the left by a certain number of units, you replace \( x \) with \( x + \text{that number} \). For shifting left 4 units, the transformation will be:
\[ f(x + 4) \]
Combining both transformations, we get:
\[ g(x) = f(x + 4) - 2 \]
Since \( f(x) = 4\sqrt{x} \), we substitute \( x + 4 \) for \( x \):
\[ f(x + 4) = 4\sqrt{x + 4} \]
Thus, applying the downward shift:
\[ g(x) = 4\sqrt{x + 4} - 2 \]
Therefore, the formula for \( g(x) \) is:
\[ g(x) = 4\sqrt{x + 4} - 2 \]
**Exercise:**
Enter the final formula for \( g(x) \):
\[ \boxed{g(x) = 4\sqrt{x + 4} - 2} \]
*Note: Enter \( \sqrt{x} \) as sqrt(x).*
[A button to "Add Work"]
[A button to proceed to the "Next Question"]
---
By following the steps outlined, students can better understand how to apply transformations to the graph of a function.
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