Question

6.

Given the graph of a function with multiple translations, write the new function rule for the translated graph.

(1 point)
The graphs below shows g(x)
, which is a translation of f(x)=x2
. Write the function rule for g(x)
. Write your answer in the form a(x−h)2+k
. Remember to use another point to solve for a
.

#1 g(x)=



Question 2
7.

The original function contains the point (10, 32) and is translated to a new function, where the original point is mapped to (5, 32). Which of the following best describes the translation in the graph?

(1 point)
Responses

The function is translated 5 units to the right and has the same slope
The function is translated 5 units to the right and has the same slope

The function is translated 5 units to the right and has a larger slope
The function is translated 5 units to the right and has a larger slope

The function is translated 5 units to the left and has a larger slope
The function is translated 5 units to the left and has a larger slope

The function is translated 5 units to the left and has the same slope

Answer 1

### Question 1

To write the function rule for \( g(x) \), we start from \( f(x) = x^2 \).

Assuming \( g(x) \) is a translation of \( f(x) \) in the form \( g(x) = a(x - h)^2 + k \), we need to identify the values of \( a \), \( h \), and \( k \).

1. **Identify Translation:**
- Determine the value of \( h \) and \( k \) by looking at how the graph of \( f(x) \) is transformed to get \( g(x) \).
- If the graph shifts to the left or right, \( h \) will be negative or positive respectively.
- If the graph shifts up or down, \( k \) will be positive or negative respectively.

2. **Find the Point for Scaling \( a \):**
- Use another point on the graph of \( g(x) \) (besides any provided) to determine \( a \).
- Substitute the coordinates of that point into the equation \( g(x) = a(x - h)^2 + k \) and solve for \( a \).

Without explicit values from a graph, I can't provide the exact function for \( g(x) \). If you provide specific points or transformations observed, I can assist with the calculations.

### Question 2

The original function has the point (10, 32) and translates to (5, 32).

1. **Change in X-Coordinate:**
- \( 10 \) to \( 5 \) is a shift to the **left** by \( 5 \) units.

2. **Change in Y-Coordinate:**
- The y-coordinate remains constant at \( 32 \), meaning the slope and vertical position of points on the graph do not change, confirming that the slope remains the **same**.

### Conclusion:
The correct description of the translation is:

**The function is translated 5 units to the left and has the same slope.**