Veronica sketched the graph of f(x) =x2 .  Then she translated it right 2 units and up 5 units to create g(x).  What is the new equation?

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To find the equation of the new function \( g(x) \), we need to apply the described transformations to the original function \( f(x) = x^2 \).

### Steps:
1. **Translation to the right by 2 units**: 
   - Translating a function to the right by \( h \) units modifies the function by replacing \( x \) with \( x - h \). Here, \( h = 2 \), so the transformation is:
   \[
   f(x - 2) = (x - 2)^2
   \]

2. **Translation up by 5 units**: 
   - Translating a function up by \( k \) units adds \( k \) to the entire function. Here, \( k = 5 \), so the final transformation is:
   \[
   g(x) = (x - 2)^2 + 5
   \]

Thus, the new equation for \( g(x) \) is:
\[
g(x) = (x - 2)^2 + 5
\]

### Answer:
The correct new equation for \( g(x) \) is \( g(x) = (x - 2)^2 + 5 \).

Do you want more details, or do you have any questions?

Here are five related questions:
1. What is the vertex of the parabola \( g(x) = (x - 2)^2 + 5 \)?
2. How does translating a graph left affect the equation?
3. What happens if the parabola \( f(x) = x^2 \) is reflected over the x-axis?
4. How would the equation change if the graph was moved down by 3 units instead?
5. What is the difference between vertical and horizontal translations?

**Tip:** When translating a graph horizontally, the transformation affects the \( x \)-coordinate, while vertical translations modify the \( y \)-coordinate.

Veronica sketched the graph of f(x) = x^2. Then she translated it right 2 units and up 5 units to create g(x). What is the new equation?

This tutorial explains how to translate the quadratic function f(x) = x^2 by moving it 2 units right and 5 units up, resulting in the new equation g(x) = (x - 2)^2 + 5. Understand function transformations and apply them to solve similar problems in algebra. Suitable for students in Grades 9-10.

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