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Let's analyze the given quadratic function:  
\[
f(x) = 2(x + 4)^2 + 3
\]

### **Step 1: Identify the Vertex**  
The given equation is in **vertex form**:  
\[
f(x) = a(x-h)^2 + k
\]  
where \( (h, k) \) is the vertex.

From \( f(x) = 2(x + 4)^2 + 3 \), we see that:  
- \( h = -4 \) (since \( x + 4 = x - (-4) \))
- \( k = 3 \)

Thus, the **vertex** is **\( (-4,3) \)**.

### **Step 2: Axis of Symmetry**  
The axis of symmetry is the vertical line that passes through the vertex.  
So, the **axis of symmetry** is **\( x = -4 \)**.

### **Step 3: Domain**  
Quadratic functions have a domain of all real numbers:  
\[
\text{Domain: } (-\infty, \infty)
\]

### **Step 4: Range**  
Since the coefficient of \( (x + 4)^2 \) is **positive (2)**, the parabola **opens upward**.  
The **minimum value** of \( f(x) \) occurs at the vertex \( (-4, 3) \), meaning the **lowest y-value is 3**.  

Thus, the **range** is:  
\[
\text{Range: } [3, \infty)
\]

### **Step 5: Y-Intercept**  
The y-intercept occurs at \( x = 0 \).  
Substituting \( x = 0 \) into the equation:

\[
f(0) = 2(0 + 4)^2 + 3
\]

\[
= 2(16) + 3 = 32 + 3 = 35
\]

So, the **y-intercept** is **\( (0,35) \)**.

---

### **Final Answers:**
- **Axis of Symmetry:** \( x = -4 \)
- **Vertex:** \( (-4,3) \)
- **Y-Intercept:** \( (0,35) \)
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( [3, \infty) \)

Would you like any further clarifications?  

### **Related Questions:**
1. How do you determine if a quadratic function has a maximum or minimum value?
2. How can we find the x-intercepts of this function?
3. What effect does changing the coefficient of \( (x+4)^2 \) have on the graph?
4. What happens to the vertex if we change \( +3 \) to \( -5 \) in the equation?
5. How do you rewrite this function in standard form?

### **Tip:**  
Always check whether the coefficient of \( x^2 \) is positive or negative to determine if the parabola opens **up** (minimum) or **down** (maximum).

Find the axis of symmetry, vertex, y-intercept, domain, and range for the quadratic function f(x) = 2(x + 4)^2 + 3.

Learn how to analyze the quadratic function f(x) = 2(x + 4)^2 + 3, including finding the axis of symmetry, vertex, y-intercept, domain, and range. This problem covers key algebraic concepts such as vertex form and the properties of quadratic functions. Suitable for students in Grades 7-9.

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