A quadratic function y, equals, f, of, xy=f(x) is plotted on a graph and the vertex of the resulting parabola is left parenthesis, 5, comma, minus, 3, right parenthesis(5,−3). What is the vertex of the function defined as g, of, x, equals, f, of, x, plus, 3, plus, 5g(x)=f(x+3)+5?

Given the function \( g(x) = f(x+3) + 5 \), let's analyze how the vertex of the original quadratic function \( f(x) \) changes.

### Step 1: Identify the effect of \( f(x+3) \)
The term \( f(x+3) \) represents a **horizontal shift** of the original function \( f(x) \). Specifically, it shifts the graph **3 units to the left**.

- The vertex of the original function \( f(x) \) is at \( (5, -3) \).
- Shifting it 3 units to the left gives the new x-coordinate of the vertex: 
  \[
  5 - 3 = 2
  \]

### Step 2: Identify the effect of adding 5
The term \( +5 \) represents a **vertical shift** of the graph **5 units upward**.

- The original y-coordinate of the vertex is \( -3 \).
- Shifting it 5 units upward gives the new y-coordinate of the vertex: 
  \[
  -3 + 5 = 2
  \]

### Step 3: Combine the shifts
After applying the shifts, the new vertex of the function \( g(x) = f(x+3) + 5 \) is:
\[
(2, 2)
\]

Thus, the vertex of the function \( g(x) \) is \( (2, 2) \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How does changing \( f(x+k) \) affect the graph of a function?
2. What happens if a constant is subtracted from the function \( f(x) \)?
3. How do vertical stretches or compressions affect the vertex of a parabola?
4. How can we determine the axis of symmetry from the vertex form of a quadratic function?
5. What is the standard form of a quadratic function?

**Tip:** The vertex form of a quadratic function \( f(x) = a(x-h)^2 + k \) makes it easy to identify the vertex \( (h, k) \) directly.

A quadratic function y=f(x) is plotted on a graph and the vertex of the resulting parabola is (5,−3). What is the vertex of the function defined as g(x)=f(x+3)+5?

Explore how the vertex of a quadratic function y=f(x) shifts when transformed to g(x)=f(x+3)+5. This problem demonstrates key algebraic concepts such as horizontal and vertical shifts and provides a step-by-step breakdown of how to calculate the new vertex. Ideal for students in Grades 9-12.

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