Answered

Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

What is the equation of the translated function, [tex]g(x)[/tex], if [tex]f(x)=x^2[/tex]?

A. [tex]g(x)=(x-4)^2+6[/tex]
B. [tex]g(x)=(x+6)^2-4[/tex]
C. [tex]g(x)=(x-6)^2-4[/tex]
D. [tex]g(x)=(x+4)^2+6[/tex]

Sagot :

GinnyAnswer
To determine the equation of the translated function [tex]\( g(x) \)[/tex] given [tex]\( f(x) = x^2 \)[/tex], let's analyze each provided option and compare them to standard transformation formulas.

### Step-by-Step Analysis:

#### Option 1: [tex]\( g(x) = (x-4)^2 + 6 \)[/tex]

1. Horizontal Translation: In [tex]\( (x-4)^2 \)[/tex], the [tex]\( x \)[/tex] is replaced by [tex]\( x-4 \)[/tex]. This translates the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally 4 units to the right.
2. Vertical Translation: The [tex]\( +6 \)[/tex] outside the squared term represents an upward translation by 6 units.

So, [tex]\( g(x) \)[/tex] in this case is [tex]\( f(x) \)[/tex] translated 4 units to the right and 6 units up.

#### Option 2: [tex]\( g(x) = (x+6)^2 - 4 \)[/tex]

1. Horizontal Translation: In [tex]\( (x+6)^2 \)[/tex], the [tex]\( x \)[/tex] is replaced by [tex]\( x+6 \)[/tex]. This translates the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally 6 units to the left.
2. Vertical Translation: The [tex]\( -4 \)[/tex] outside the squared term represents a downward translation by 4 units.

So, [tex]\( g(x) \)[/tex] in this case is [tex]\( f(x) \)[/tex] translated 6 units to the left and 4 units down.

#### Option 3: [tex]\( g(x) = (x-6)^2 - 4 \)[/tex]

1. Horizontal Translation: In [tex]\( (x-6)^2 \)[/tex], the [tex]\( x \)[/tex] is replaced by [tex]\( x-6 \)[/tex]. This translates the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally 6 units to the right.
2. Vertical Translation: The [tex]\( -4 \)[/tex] outside the squared term represents a downward translation by 4 units.

So, [tex]\( g(x) \)[/tex] in this case is [tex]\( f(x) \)[/tex] translated 6 units to the right and 4 units down.

#### Option 4: [tex]\( g(x) = (x+4)^2 + 6 \)[/tex]

1. Horizontal Translation: In [tex]\( (x+4)^2 \)[/tex], the [tex]\( x \)[/tex] is replaced by [tex]\( x+4 \)[/tex]. This translates the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally 4 units to the left.
2. Vertical Translation: The [tex]\( +6 \)[/tex] outside the squared term represents an upward translation by 6 units.

So, [tex]\( g(x) \)[/tex] in this case is [tex]\( f(x) \)[/tex] translated 4 units to the left and 6 units up.

### Conclusion

Among the given options:
- The transformation that translates [tex]\( f(x) = x^2 \)[/tex] horizontally 4 units to the right and vertically 6 units up is found in Option 1.

Thus, the equation of the translated function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = (x-4)^2 + 6 \][/tex]

So, the correct answer is:
[tex]\[ g(x) = (x-4)^2 + 6 \][/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.