Question
What are the order of combinations to transform f(x) = x^2 to f(x+2) - 5?
Answers
Answered by
GPT-4o mini
To transform the function \( f(x) = x^2 \) into \( f(x+2) - 5 \), we will apply two main transformations in order: a horizontal shift and a vertical shift.
### Step 1: Horizontal Shift
First, we need to shift the function \( f(x) = x^2 \) horizontally to the left by 2 units. This is done by replacing \( x \) with \( x+2 \):
\[
f(x + 2) = (x + 2)^2
\]
### Step 2: Vertical Shift
Next, we need to shift the entire function downwards by 5 units. This is done by subtracting 5 from the function we just found:
\[
f(x + 2) - 5 = (x + 2)^2 - 5
\]
### Final Expression
So after performing both transformations, we arrive at the final expression:
\[
f(x + 2) - 5 = (x + 2)^2 - 5
\]
### Summary of the Combinations
1. **Horizontal Shift:** Replace \( x \) with \( x+2 \) to get \( (x + 2)^2 \).
2. **Vertical Shift:** Subtract 5 from the resulting function to get \( (x + 2)^2 - 5 \).
Thus, the order of transformations is: **horizontal shift**, then **vertical shift**.
### Step 1: Horizontal Shift
First, we need to shift the function \( f(x) = x^2 \) horizontally to the left by 2 units. This is done by replacing \( x \) with \( x+2 \):
\[
f(x + 2) = (x + 2)^2
\]
### Step 2: Vertical Shift
Next, we need to shift the entire function downwards by 5 units. This is done by subtracting 5 from the function we just found:
\[
f(x + 2) - 5 = (x + 2)^2 - 5
\]
### Final Expression
So after performing both transformations, we arrive at the final expression:
\[
f(x + 2) - 5 = (x + 2)^2 - 5
\]
### Summary of the Combinations
1. **Horizontal Shift:** Replace \( x \) with \( x+2 \) to get \( (x + 2)^2 \).
2. **Vertical Shift:** Subtract 5 from the resulting function to get \( (x + 2)^2 - 5 \).
Thus, the order of transformations is: **horizontal shift**, then **vertical shift**.