A quadratic function y, equals, f, of, x*y*=*f*(*x*) is plotted on a graph and the vertex of the resulting parabola is left parenthesis, 4, comma, minus, 6, right parenthesis(4,−6). What is the vertex of the function defined as g, of, x, equals, f, of, x, plus, 2, plus, 4*g*(*x*)=*f*(*x*+2)+4?

To find the vertex of the function \( g(x) = f(x+2) + 4 \), we need to understand how the transformation affects the vertex of the original function \( f(x) \).

### Given:
- The original function \( f(x) \) has a vertex at \( (4, -6) \).
- The transformed function is \( g(x) = f(x+2) + 4 \).

### Step 1: Horizontal Shift
The term \( x + 2 \) inside the function \( f(x+2) \) represents a horizontal shift to the left by 2 units. 

Therefore, the vertex of \( f(x) \), originally at \( (4, -6) \), moves to the point \( (4 - 2, -6) = (2, -6) \).

### Step 2: Vertical Shift
The term \( +4 \) outside the function \( f(x+2) \) represents a vertical shift upwards by 4 units. 

So, the vertex at \( (2, -6) \) will move up by 4 units, resulting in a new vertex at \( (2, -6 + 4) = (2, -2) \).

### Final Answer:
The vertex of the function \( g(x) = f(x+2) + 4 \) is \( \boxed{(2, -2)} \).

Would you like more details or have any further questions?

Here are five related questions:
1. How does a horizontal shift affect the graph of a quadratic function?
2. What is the effect of a vertical shift on the vertex of a quadratic function?
3. Can you find the vertex of a function if it undergoes both horizontal and vertical shifts?
4. How does the vertex form of a quadratic function help in identifying shifts?
5. What would be the vertex if the transformation was \( g(x) = f(x-3) - 5 \)?

**Tip:** When working with function transformations, always apply horizontal shifts before vertical ones.

A quadratic function y = f(x) is plotted on a graph and the vertex of the resulting parabola is (4, -6). What is the vertex of the function defined as g(x) = f(x + 2) + 4?

Explore how to find the vertex of a transformed quadratic function g(x) = f(x + 2) + 4, given the original vertex at (4, -6). This problem involves understanding horizontal and vertical shifts and is suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation